CHAPTER 3
Amenability
As seen in Section 1.2, the basic theory of unitary representations is rooted in a simple dichotomy: such a representation is either weakly mixing or has a nonzero finitedimensional subrepresentation. Finite-dimensional unitary representations of a group G
carry a special algebraic and geometric structure that results from the fact that the unitary
n × n matrices form a compact Lie group, on which we can view G as acting by left
multiplication. Useful as this might be, many groups do not even have finite-dimensional
unitary representations except for direct sums of trivial representations. However, what
is important for many purposes is whether a unitary representation π : G → B(H)
displays a less rigid version of “finiteness” that merely requires the existence of nonzero
finite-dimensional subspaces of H which are almost invariant in a suitable sense (see Definition 3.11). We can then distinguish two properties for a group G, with the convention
that we are talking about nonzero subspaces:
(i) every weakly mixing unitary representation of G has almost invariant finitedimensional subspaces,
(ii) no weakly mixing unitary representation of G has almost invariant finite-dimensional
subspaces.
The first is a characterization of amenability and the second of property (T). Not surprisingly given the above formulations, property (T) has strong rigidity consequences for
p.m.p. actions and is relatively difficult to establish in examples, while p.m.p. actions of
amenable groups are distinguished by a detailed structure theory based on finite approximation. The latter structure theory takes on different technical expressions depending on
whether we are interested in the relation of conjugacy or the much weaker relation of orbit
equivalence, both of which are fundamental for a complete understanding of amenability. For questions where conjugacy is the operative relation, we have the machinery of
quasitilings (Section 3.5) and the Rokhlin lemma (Section 3.6). Orbit equivalence on
the other hand is handled by the Connes-Feldman-Weiss theorem (Section 3.7) and the
Dye and Ornstein-Weiss theorems (Section 3.8). The Ornstein-Weiss theorem is a consequence of the theorems of Connes-Feldman-Weiss and Dye and asserts that any two
ergodic p.m.p. actions of an amenable group on an atomless probability space are orbit
equivalent. The strategy in passing from approximate invariance to finite approximation
91
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3. AMENABILITY
in our proofs of both the Rokhlin lemma and the Connes-Feldman-Weiss theorem is to
first establish a local version of the conclusion on parts of the space and then leverage this
locality by a Zorn’s lemma maximality argument to show that the whole space can almost
completely be filled up with local finite approximations.
The basic definitions and characterizations of amenability are treated in Section 3.1,
while Section 3.2 characterizes amenability in terms of unitary representations. Section 3.3 contains the mean ergodic theorem and Section 3.4 the pointwise ergodic theorems. Property (T) is treated separately in Chapter 4.
Note that although our blanket assumption throughout the book is that G is always a
countable discrete group, the results and arguments of Sections 3.1 and 3.2 are equally
valid for uncountable discrete groups granted that we replace sequences by nets in the
appropriate places, as for instance in the Følner condition.
3.1. Basic theory
A mean for G on ℓ∞ (G) is a unital positive linear functional σ : ℓ∞ (G) → C (unital
means that σ(1) = 1). On general unital C∗ -algebras such a linear functional is called a
state. A mean σ is automatically bounded with norm one, for if f ∈ ℓ∞ (G) then there is
a z ∈ T such that |σ(f )| = zσ(f ) and hence
|σ(f )| = σ(zf ) = re σ(zf ) = σ(re zf ) ≤ k re zf k ≤ kzf k = kf k.
The mean σ is left invariant if σ(s · f ) = σ(f ) for all s ∈ G and f ∈ ℓ∞ (G) where
(s · f )(t) = f (s−1 t) for all t ∈ G.
D EFINITION 3.1. The group G is said to be amenable if there is a left invariant mean
on ℓ∞ (G).
P ROPOSITION 3.2.
(i) Every finite group is amenable.
(ii) If G is amenable then so are all of its quotients and subgroups.
(iii) If N is a normal subgroup of G such that N and G/N are amenable, then G is
amenable.
(iv) If G and H are amenable then so is G × H.
(v) If G is the union of an increasing sequence of amenable subgroups, then G is
amenable.
(vi) If every finitely generated subgroup of G is amenable, then G is amenable.
P
P ROOF. (i). If G is a finite group then f 7→ |G|−1 s∈G f (s) defines a left invariant
mean on ℓ∞ (G).
3.1. BASIC THEORY
93
(ii). If H is a quotient of G, then ℓ∞ (H) embeds into ℓ∞ (G) via composition with the
quotient map G → H. By composing with this embedding, any left invariant mean for G
on ℓ∞ (G) yields one for H on ℓ∞ (H).
Suppose now that we are given a subgroup H of G. Take a set R of representatives for
the right cosets of H in G and define the unital positive linear map ϕ : ℓ∞ (H) → ℓ∞ (G)
by ϕ(f )(st) = f (s) for all s ∈ H and t ∈ R. If σ is a left invariant mean for G on ℓ∞ (G),
then σ ◦ ϕ is a mean on ℓ∞ (H), and it is left invariant for H since ϕ(s · f ) = s · ϕ(f ) for
all f ∈ ℓ∞ (H) and s ∈ H.
(iii). Let σ be a left invariant mean for N on ℓ∞ (N ) and ω a left invariant mean for
G/N on ℓ∞ (G/N ). Take a set R of coset representatives in G for the elements of G/N .
Define a unital positive linear map ϕ : ℓ∞ (G) → ℓ∞ (G/N ) by setting ϕ(f )(sN ) =
σ((s−1 f )|N ) for every f ∈ ℓ∞ (G) and s ∈ G, which is well defined since σ is left
invariant for N . Then for all f ∈ ℓ∞ (G) and s, t ∈ G we have
ϕ(sf )(tN ) = σ((t−1 sf )|N ) = ϕ(f )(s−1 tN ) = (sN · ϕ(f ))(tN )
and hence ω(ϕ(sf )) = ω(sN · ϕ(f )) = ω(ϕ(f )), showing that ω ◦ ϕ is a left invariant
mean for G.
(iv). Let ι be the embedding s 7→ (e, s) of H into G × H. Then ι(H) is a normal
subgroup of G × H with (G × H)/ι(H) ∼
= G, so that we can apply (iii).
(v). Let H1 ⊆ H2 ⊆ . . . be an increasing sequence of amenable subgroups of G
whose union is equal to G. Then for every n there is a left invariant mean σn for Hn
on ℓ∞ (Hn ). Letting ϕn : ℓ∞ (G) → ℓ∞ (Hn ) be the map given by composition with the
inclusion Hn ֒→ G, it is then readily checked that any weak∗ limit point of the sequence
{σn ◦ ϕn } is a left invariant mean for G on ℓ∞ (G).
(vi). Take an enumeration s1 , s2 , . . . of the elements of G. Then for each n ∈ N
the subgroup generated by s1 , . . . , sn is amenable, and since G is the increasing union of
these subgroups we conclude by (v) that G is amenable.
By combining (i) and (v) above we can construct many infinite amenable groups by
taking direct limits of finite groups. Such groups are called locally finite. As an example,
consider the group of permutations of N which fix all but finitely many elements, which
can be expressed as the increasing union over n ∈ N of the groups of permutations fixing
the elements {n + 1, n + 2, . . . }.
The property of amenability is remarkable for its wide and ever-growing variety of formulations. We next establish some fundamental characterizations of amenability, ranging
from the Følner property at the formally strongest extreme to nonparadoxicality at the
formally weakest. The Følner property is useful for exhibiting amenable groups which
are not locally finite, while paradoxicality is useful for exhibiting nonamenable groups.
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3. AMENABILITY
D EFINITION 3.3. (i) A sequence {Fn } of nonempty finite subsets of G is called a
(left) Følner sequence if |sFn ∆Fn |/|Fn | → 0 as n → ∞ for every s ∈ G. It is called
a two-sided Følner sequence if both |sFn ∆Fn |/|Fn | → 0 and |Fn s∆Fn |/|Fn | → 0 as
n → ∞ for every s ∈ G.
(ii) We say that G satisfies Reiter’s property if there is a sequence {fn } of nonnegative
real-valued functions in ℓ1 (G) of norm one such that ksfn − fn k1 → 0 as n → ∞ for
every s ∈ G.
(iii) Two sets C, D ⊆ G are said to be equidecomposable, written C ∼ D, if there
exist sets C1 , . . . , Cn ⊆ C and s1 , . . . , sn ∈ G such that C = C1 ⊔ · · · ⊔ Cn and D =
s1 C1 ⊔ · · · ⊔ sn Cn . This defines an equivalence relation on the collection of subsets of G.
We say that G is paradoxical if there exist disjoint sets C, D ⊆ G such that C ∼ D ∼ G.
One can show by a Schröder-Bernstein argument that if G is paradoxical then we
may assume that the C and D in the definition of paradoxicality actually partition G, in
which case we speak of a paradoxical decomposition of G. Indeed suppose that C and
D are disjoint subsets of G such that C ∼ D ∼ G. Take sets E1 , . . . , En ⊆ G and
s1 , . . . , sn ∈ G such that G = E1 ⊔ · · · ⊔ En and D = s1 E1 ⊔ · · · ⊔ sn En . Let f : G → D
be the bijection given on Ek by x 7→ sk x. Recursively define the pairwise disjoint sets
F
A0 = C and Aj+1 = f (Aj ) for j ≥ 0, and set A = ∞
j=0 Aj . Then G\A = (G\C)\f (A)
and consequently
G\C = (G\A) ⊔ f (A) ∼ (G\A) ⊔ A = G,
so that C and G\C provide a paradoxical decomposition of G.
T HEOREM 3.4. The following are equivalent.
(i) G is amenable,
(ii) there is a G-invariant finitely additive measure µ on G with µ(G) = 1,
(iii) every continuous action of G on a nonempty compact Hausdorff space admits
an invariant regular Borel probability measure,
(iv) G has Reiter’s property,
(v) G admits a Følner sequence,
(vi) G is not paradoxical.
P ROOF. (i)⇒(ii). Given a left invariant mean σ on ℓ∞ (G) we can set µ(C) = σ(1C )
for every C ⊆ G to obtain a finitely additive measure µ of the desired type.
(ii)⇒(vi). Let µ be a G-invariant finitely additive measure µ on G with µ(G) = 1. If
G is paradoxical then there exist disjoint sets C, D ⊆ G such that C ∼ D ∼ G, in which
case 1 ≥ µ(C) + µ(D) = µ(G) + µ(G) = 2, a contradiction.
(vi)⇒(v). Suppose that G does not admit a Følner sequence. Then we can find a λ > 1
and a nonempty finite set S ⊆ G such that |SF | ≥ λ|F | for every finite set F ⊆ G (we
3.1. BASIC THEORY
95
may assume e ∈ S in order to get this inequality). This failure of approximate invariance
can be geometrically amplified as follows so as to produce a larger S that at least doubles
the size of every F . Choose an n ∈ N such that λn ≥ 2. Assuming that e ∈ S as we may,
for every finite set F ⊆ G we have |S k F | = |S(S k−1 F )| ≥ λ|S k−1 F | for k = 1, . . . , n
and hence by induction
|S n F | ≥ λn |F | ≥ 2|F |.
By replacing S with S n we may therefore assume that |SF | ≥ 2|F | for all finite sets
F ⊆ G.
From these local size-doublings we will extract a global two-to-one matching that will
yield the desired paradoxicality. Let Ω be the collection of all families {Ax }x∈G×{1,2} of
finite subsets of G indexed by G × {1, 2} such that
S
(a) Ax ≥ |K| for every finite set K ⊆ G × {1, 2}, and
x∈K
(b) Ax ⊆ St for every x = (t, i) ∈ G × {1, 2}.
Note that {St}(t,i)∈G×{1,2} is an element of Ω, for given a finite set K ⊆ G × {1, 2} and
writing it as (K1 × {1}) ∪ (K2 × {2}) we have
[
= |S(K1 ∪ K2 )| ≥ 2|K1 ∪ K2 | ≥ |K|.
St
(t,i)∈K
Thus Ω is nonempty. Ordering Ω by indexwise inclusion, we observe that every chain
in Ω has a lower bound obtained by taking indexwise intersections. It follows by Zorn’s
Lemma that Ω has a minimal element {Mx }.
We claim that |Mx | = 1 for all x ∈ G × {1, 2}. Suppose not. Since each Mx is
nonempty this means that we can find an x0 for which Mx0 contains two distinct elements
t1 and t2 . Then by minimality we can find for each i = 1, 2 a finite set Ki ⊆ G × {1, 2}
S
not containing x0 such that the set Ri = (Mx0 \{ti }) ∪ x∈Ki Mx satisfies |Ri | ≤ |Ki |. It
follows that
|K1 | + |K2 | ≥ |R1 | + |R2 | = |R1 ∪ R2 | + |R1 ∩ R2 |
[
[
Mx ≥ Mx0 ∪
Mx + x∈K1 ∪K2
x∈K1 ∩K2
≥ 1 + |K1 ∪ K2 | + |K1 ∩ K2 | = 1 + |K1 | + |K2 |,
giving a contradiction which verifies the claim.
Note that the singletons Mx for x ∈ G × {1, 2} must be pairwise disjoint by virtue of
{Mx } being an element of Ω. For each s ∈ S define the sets
Cs = t ∈ G : st ∈ M(t,1) , Ds = t ∈ G : st ∈ M(t,2) .
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3. AMENABILITY
Then each of {Cs : s ∈ S} and {Ds : s ∈ S} is a partition of G and {sCs : s ∈
S} ∪ {sDs : s ∈ S} is a disjoint family, showing that G is paradoxical.
(v)⇒(iv). Given a Følner sequence {Fn }, the sequence {|Fn |−1 1Fn } in ℓ1 (G)+ witnesses Reiter’s property.
(iv)⇒(iii). Let {fn } be a sequence in ℓ1 (G)+ witnessing Reiter’s property. Let G y
X be a continuous action on a nonempty compact Hausdorff space. Choose a point x ∈
X. Writing δsx for the point mass at sx, it follows that every weak∗ cluster point of
P
the net { s∈G fn (s)δsx } within the weak∗ compact space of all regular Borel probability
measures on X is G-invariant.
(iii)⇒(i). Every invariant regular Borel probability measure for the action G y βG
gives rise via integration to a G-invariant unital positive linear functional on C(βG),
which is G-equivariantly isomorphic to ℓ∞ (G) as a C ∗ -algebra.
E XAMPLE 3.5. The group Z is amenable, since the sets {1, . . . , n} for n ∈ N form a
Følner sequence. More generally, every Abelian G is amenable. To verify this, it suffices
by Proposition 3.2 to assume that G is finitely generated. By the fundamental theorem for
finitely generated Abelian groups, there are integers r ≥ 0 and k1 , . . . , km ≥ 2 such that
we can view G up to isomorphism as Zr ⊕ (Z/k1 Z) ⊕ · · · ⊕ (Z/km Z). Then for n ∈ N
the sets {1, . . . , n}r ⊕ (Z/k1 Z) ⊕ · · · ⊕ (Z/km Z) form a Følner sequence.
E XAMPLE 3.6. It follows from Example 3.5 and Proposition 3.2(iii) that every solvable G is amenable. In particular, every nilpotent G is amenable.
E XAMPLE 3.7. Let r be an integer greater than 1. Then the free group Fr of rank r
is not amenable. Since Fr contains F2 as a subgroup, by Proposition 3.2(ii) and Theorem 3.4 it is enough to verify that F2 is paradoxical. We will in fact exhibit a paradoxical decomposition of F2 . Write the standard generating set of F2 as S = {a, b}. For
s ∈ S ∪ S −1 write Ws for the set of all reduced words in S ∪ S −1 beginning with s. Set
C1 = Wb ∪ {e, b−1 , b−2 , . . . }, C2 = Wb−1 \{b−1 , b−2 , . . . }, D1 = Wa , and D2 = Wa−1 .
Then {C1 , C2 , D1 , D2 } is a partition of G, and F2 = C1 ⊔ bC2 = D1 ⊔ aD2 .
The definitions of amenability, (left) Følner sequence, Reiter’s property, equidecomposability, and paradoxicality in Definitions 3.1 and 3.3, as well as all of the conditions
in Theorem 3.4 except for (iii), are framed in terms of the left action of G on itself, and
make sense more generally for any action G y Y of G on a nonempty (discrete) set
Y . We merely need to substitute Y for G wherever G appears in its role as a space. We
thus say for example that such an action G y Y is amenable if there is a left invariant
mean on ℓ∞ (Y ), by which we mean a positive linear functional σ : ℓ∞ (Y ) → C such that
σ(sf ) = σ(f ) for all s ∈ G and f ∈ ℓ∞ (Y ), where sf (y) = f (s−1 y) for all y ∈ Y . Also,
a (left) Følner sequence is a sequence {Fn } of nonempty finite subsets of Y such that
3.2. AMENABILITY AND UNITARY REPRESENTATIONS
97
|sFn ∆Fn |/|Fn | → 0 as n → ∞ for every s ∈ G. The proof of Theorem 3.4 then works
verbatim for the action G y Y once we replace G in its spatial role everywhere by Y ,
except that we must skip over (iii) and argue (iv)⇒(i) directly by taking a weak∗ cluster
point in ℓ∞ (Y )∗ of a sequence {fn } of norm-one nonnegative real-valued functions in
ℓ1 (Y ) satisfying ksfn − fn k1 → 0 as n → ∞ for all s ∈ G.
T HEOREM 3.8. Let G y Y be an action on a nonempty set. The following are
equivalent.
(i) the action is amenable,
(ii) there is a G-invariant finitely additive measure µ on Y with µ(Y ) = 1,
(iii) the action has Reiter’s property,
(iv) the action admits a Følner sequence,
(v) Y is not G-paradoxical.
One virtue of this more general form of Theorem 3.4 is that we can use it as follows to
show that an amenable G admits a two-sided Følner sequence, a fact which will be used
in the proof of Theorem ??.
L EMMA 3.9. Suppose that G is amenable. Then every action of G on a nonempty set
is amenable.
P ROOF. Let σ be a left invariant mean on ℓ∞ (G), and let G y Y be an action on a
nonempty set. Pick a y ∈ Y and define a map ϕ : G → Y by s 7→ sy. Define the unital
positive linear map Φ : ℓ∞ (Y ) → ℓ∞ (G) by Φ(f ) = f ◦ ϕ. Then the composition σ ◦ Φ
is a left invariant mean on ℓ∞ (Y ), showing that the action is amenable.
T HEOREM 3.10. Suppose that G is amenable. Then it admits a two-sided Følner
sequence.
P ROOF. Consider the action G × G y G given by (s, t) · u = sut−1 . By Proposition 3.2(iv) the group G × G is amenable, and so by Lemma 3.9 the action is amenable.
It follows by Theorem 3.8 that the action admits a Følner sequence, i.e., a sequence
{Fn } of nonempty finite subsets of G such that limn→∞ |sFn t−1 ∆Fn |/|Fn | = 0 for all
(s, t) ∈ G × G. This sequence is then a two-sided Følner sequence for G.
3.2. Amenability and unitary representations
Here we characterize amenable groups in terms of their unitary representations. Central to the discussion is the conjugation representation on HS(H) associated to a unitary
representation π : G → B(H), defined by T 7→ π(s)T π(s)∗ for T ∈ HS(H) and s ∈ G.
This is unitarily equivalent to π ⊗ π̄ in a canonical way (Section 0.8), and so for notational convenience we will simply write π ⊗ π̄ keeping this unitary equivalence implicitly
in mind.
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3. AMENABILITY
We begin by defining the unitary representation analogue of the Følner property.
D EFINITION 3.11. A unitary representation π : G → B(H) has almost invariant
finite-dimensional subspaces if for every finite set F ⊆ G and ε > 0 there is a nonzero
finite rank projection P ∈ B(H) such that kπ(s)P π(s)∗ − P k2 ≤ εkP k2 for all s ∈ F .
D EFINITION 3.12. A unitary representation π : G → B(H) has almost invariant
vectors if for every finite set F ⊆ G and ε > 0 there is a unit vector ξ ∈ H such that
kπ(s)ξ − ξk2 < ε for all s ∈ F .
We next aim to show in Proposition 3.17 that we can equivalently ask for P in Definition 3.11 to be any nonzero Hilbert-Schmidt operator, and also that this property is
equivalent to π ⊗ π̄ having almost invariant vectors in the above sense. For this we will
need some inequalities (Lemmas 3.13, 3.15, and 3.16) to enable us to apply Reiter’s property within the context of Hilbert-Schmidt operators.
For a self-adjoint operator h on a Hilbert space H and a Borel set A ⊆ R, we write
EA (h) for the spectral projection of h over A [76, Sect. 5.2][105, Sect. 2.5]. If H decomposes into a direct sum of eigenspaces for h (which happens if h is compact, and in
particular if h is Hilbert-Schmidt), then EA (h) is the projection onto the direct sum of
those eigenspaces whose eigenvalue lies in A. If X is a σ-finite measure space and h
is an element of L∞ (X) acting by multiplication on L2 (X), then EA (h) is the indicator
function of the set {x ∈ X : h(x) ∈ A}. For r ∈ R we abbreviate E[r,∞) (h) to Er (h).
L EMMA 3.13 (Powers-Størmer inequality). Let h and k be positive Hilbert-Schmidt
operators on a Hilbert space H. Then
kh − kk22 ≤ kh2 − k 2 k1 .
P ROOF. Write p for the spectral projection of the self-adjoint operator h − k over the
interval [0, ∞). Set p⊥ = 1 − p. Using the fact that p commutes with h − k, we have
Tr((h2 − k 2 )p) − Tr((h − k)2 p) = Tr(pk(h − k)p) + Tr(p(h − k)kp)
= Tr(pkp(h − k)) + Tr((h − k)pkp)
= 2Tr(kp(h − k)p)
√
√
= 2Tr( k(p(h − k)p) k)
and this last quantity is nonnegative because p(h − k)p is a positive operator. Similarly,
Tr((k 2 − h2 )p⊥ ) − Tr((k − h)2 p⊥ ) ≥ 0. Therefore
kh2 − k 2 k1 = Tr(|h2 − k 2 |p) + Tr(|k 2 − h2 |p⊥ )
≥ Tr((h2 − k 2 )p) + Tr((k 2 − h2 )p⊥ )
≥ Tr((h − k)2 p) + Tr((k − h)2 p⊥ )
3.2. AMENABILITY AND UNITARY REPRESENTATIONS
= Tr((h − k)2 (p + p⊥ )) = kh − kk22 .
99
L EMMA 3.14. Let h and k be positive Hilbert-Schmidt operators on a Hilbert space
H. Set X = (R+ × R+ )\{(0, 0)} where R+ = [0, ∞), and define the functions H and K
on X by H(x, y) = x and K(x, y) = y. Then there is a σ-finite Borel measure µ on X
such that
(i) khk2 = kHk2 , kkk2 = kKk2 , and kh − kk2 = kH − Kk2 ,
(ii) for each r ∈ (0, ∞) the spectral projections Er (H) and Er (K) over [r, ∞) are
square-integrable and kEr (h) − Er (k)k2 = kEr (H) − Er (K)k2 .
P ROOF. For any ε > 0 and Borel set A ⊆ (ε, ∞) we have EA (h) ≤ ε−2 h2 , so that the
positive operator EA (h) is Hilbert-Schmidt. Then for any Borel set B p
⊆ R+ the quantity
p
Tr(EA (h)EB (k)) is finite, and it lies in R+ since it can be written as Tr( EA (h)EB (k) EA (h)).
Interchanging the roles of h and k, we similarly see that for any Borel sets A ⊆ R+
and B ⊆ (ε, ∞) the quantity Tr(EA (h)EB (k)) lies in R+ . Declaring µ0 (A × B) to be
Tr(EA (h)EB (k)) in both cases, we obtain a σ-finite premeasure µ0 on the algebra of
subsets of X generated by the Borel rectangles which are contained in a set of the form
[ε, ∞) × R+ or R+ × [ε, ∞) for some ε > 0. By Carathéodory’s extension theorem, this
premeasure extends uniquely to a σ-finite Borel measure µ on X, which is evidently finite on compact sets. Since h and k are Hilbert-Schmidt and hence compact, their spectra
σ(h) and σ(k) are countable sets consisting of eigenvalues and possibly also 0, which is
the unique cluster point in the case that H is infinite-dimensional. Thus µ, being supported
on (σ(h) × σ(k))\{(0, 0)}, is atomic. Letting λ range through the nonzero eigenvalues
of h and ζ through the nonzero eigenvalues of k, for any square-summable collections of
nonnegative real numbers {cλ } and {dζ } we have
2
X
X
cλ E{λ} (h) −
dζ E{ζ} (k)
λ
=
X
λ
=
X
λ
ζ
c2λ Tr(E{λ} (h)) +
X
c2λ µ({λ} × R+ ) +
ζ
2
d2ζ Tr(E{ζ} (k)) − 2
X
ζ
ζ
λ,ζ
d2ζ µ(R+ × {ζ}) − 2
2
X
X
.
=
c
1
−
d
1
λ
ζ
{λ}×R
R
×{ζ}
+
+
λ
X
cλ dζ Tr(E{λ} (h)E{ζ} (k))
X
λ,ζ
cλ dζ µ({λ} × {ζ})
2
From this we can deduce both (i) and (ii) by making appropriate choices of cλ and dζ .
For (ii), given an r > 0 we choose cλ to be 1 if λ ≥ r and 0 otherwise so that Er (h) =
P
P
λ cλ 1{λ}×R+ , and make a similar choice for the dζ . For (i),
λ cλ E{λ} (h) and Er (H) =
choose cλ = λ and dζ = 0, then cλ = 0 and dζ = ζ, and finally cλ = λ and dζ = ζ.
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3. AMENABILITY
L EMMA 3.15. Let h and k be positive Hilbert-Schmidt operators on a Hilbert space
H. Then
Z ∞
kE√r (h) − E√r (k)k22 dr ≤ (khk2 + kkk2 )kh − kk2 .
0
P ROOF. By Lemma 3.14, there are a locally compact metrizable space X and a σfinite Borel measure µ on X such that we may assume h and k to be square-integrable
functions in C0 (X) acting on L2 (X, µ) by multiplication. Then for every r > 0 the multiplication operators Er (h2 ) and Er (k 2 ) are commuting projections, so that the operator
|Er (h2 ) − Er (k 2 )| is a projection and hence kEr (h2 ) − Er (k 2 )k22 = kEr (h2 ) − Er (k 2 )k1 .
Thus, applying Fubini’s theorem,
Z ∞
Z ∞
2
√
√
kEr (h2 ) − Er (k 2 )k22 dr
kE r (h) − E r (k)k2 dr =
0
Z0 ∞ Z
|Er (h2 )(x) − Er (k 2 )(x)| dµ(x) dr
=
X
Z0 Z ∞
|Er (h2 )(x) − Er (k 2 )(x)| dr dµ(x)
=
ZX 0
|h2 (x) − k 2 (x)| dµ(x)
=
ZX
|h(x) + k(x)||h(x) − k(x)| dµ(x)
=
X
≤ kh + kk2 kh − kk2
≤ (khk2 + kkk2 )kh − kk2
where the second last inequality is the Cauchy-Schwarz inequality.
L EMMA 3.16. Let h and k be Hilbert-Schmidt operators on a Hilbert space. Then
kh∗ h − k ∗ kk1 ≤ (khk2 + kkk2 )kh − kk2 .
P ROOF. Since h∗ h − k ∗ k is a self-adjoint operator, it has a spectral projection p over
the interval [0, ∞), and so writing p⊥ = 1 − p and using the Cauchy-Schwarz inequality
we obtain
kh∗ h − k ∗ kk1 = Tr(|h∗ h − k ∗ k|)
= Tr((h∗ h − k ∗ k)(p − p⊥ ))
≤ |Tr(h∗ (h − k)(p − p⊥ ))| + |Tr((h − k)∗ k(p − p⊥ ))|
≤ (kh(p − p⊥ )k2 + kk(p − p⊥ )k2 )kh − kk2
= (khk2 + kkk2 )kh − kk2 .
3.2. AMENABILITY AND UNITARY REPRESENTATIONS
101
P ROPOSITION 3.17. For a unitary representation π : G → B(H), the following are
equivalent.
(i) π ⊗ π̄ has almost invariant vectors,
(ii) for every finite set F ⊆ G and ε > 0 there is a T ∈ HS(H) of Hilbert-Schmidt
norm one such that kπ(s)T π(s)∗ − T k2 < ε for all s ∈ F ,
(iii) π has almost invariant finite-dimensional subspaces.
P ROOF. The implications (i)⇔(ii) are immediate from the unitary equivalence of π ⊗
π̄ with the conjugation representation on HS(H), and (iii)⇒(ii) is trivial.
Let us now prove (ii)⇒(iii). By (ii), given a finite set F ⊆ G and ε > 0 there exists an
operator Q on H of Hilbert-Schmidt norm one such that kπ(s)Qπ(s)∗ −Qk2 < ε4 /(8|F |2 )
√
for all s ∈ F . Set T = |Q| = Q∗ Q. Then T is a positive operator of Hilbert-Schmidt
norm one, and applying Lemmas 3.13 and 3.16 we obtain, for every s ∈ F ,
kπ(s)T π(s)∗ − T k22 ≤ kπ(s)Q∗ Qπ(s)∗ − Q∗ Qk1
≤ 2kπ(s)Qπ(s)∗ − Qk2 <
ε4
.
4|F |2
It follows by Lemma 3.15 that for every s ∈ F we have
Z ∞
ε2
.
kE√r (π(s)T π(s)∗ ) − E√r (T )k22 dr ≤ 2kπ(s)T π(s)∗ − T k2 ≤
|F |
0
Since
Z
∞
0
kE√r (T )k22
dr =
Z
∞
0
kEr (T 2 )k22 dr = kT 2 k1 = kT k22 = 1
and E√r (π(s)T π(s)∗ ) = π(s)E√r (T )π(s)∗ for every r ≥ 0, we infer that
Z ∞
Z ∞X
∗
2
ε2 kE√r (T )k22 dr.
kπ(s)E√r (T )π(s) − E√r (T )k2 dr ≤
0
s∈F
0
Consequently there exists an r > 0 with E√r (T ) 6= 0 such that kπ(s)E√r (T )π(s)∗ −
E√r (T )k22 ≤ ε2 kE√r (T )k22 for every s ∈ F , showing that π has almost invariant finitedimensional subspaces.
The following fact will be useful elsewhere and so we separate it out as a lemma.
L EMMA 3.18. Let π : G → B(H) be a unitary representation, and suppose that for
some index set I the representation π ⊕I on H⊕I has almost invariant vectors. Then π has
almost invariant vectors.
102
3. AMENABILITY
P ROOF. Observe that, for every finite set F ⊆ G, ε > 0, and unit vector f = (fi ) in
H , the inequality
XX
X
X
kπ(s)fi − fi k22 =
kπ ⊕I (s)f − f k22 < ε =
εkfi k22
⊕I
i∈I s∈F
P
s∈F
i∈I
implies that s∈F kπ(s)fi − fi k22 < εkfi k22 for some i. From this we deduce that π has
almost invariant vectors.
T HEOREM 3.19. The following are equivalent.
(i) G is amenable,
(ii) π ⊗ π̄ has almost invariant vectors for every unitary representation π of G,
(iii) every weakly mixing unitary representation of G has almost invariant finitedimensional subspaces,
(iv) λG has almost invariant vectors.
P ROOF. (i)⇒(ii). Let π : G → B(H) be a unitary representation of G. Let F be a
finite subset of G and ε > 0. Our assumption that G is amenable means that it has Reiter’s
property by Theorem 3.4, and so there is a nonnegative norm-one function f ∈ ℓ1 (G) such
that ksf − f k1 < ε for all s ∈ F . Choose a positive operator R on H of trace class norm
one and set
X
Q=
f (t)π(t)Rπ(t)−1 ,
t∈G
√
which is a positive operator of trace class norm one. Now set T = Q, which has
Hilbert-Schmidt norm one. Then for every s ∈ F we have, using Lemma 3.13,
kπ(s)T π(s)−1 − T k22 ≤ kπ(s)Qπ(s)−1 − Qk1
X
−1
−1
f
(s
t)
−
f
(t)
π(t)Rπ(t)
=
1
t∈G
X
≤
|f (s−1 t) − f (t)|
t∈G
= ksf − f k1 < ε.
It follows by Proposition 3.17 that π ⊗ π̄ has almost invariant vectors.
(ii)⇔(iii). Apply Proposition 3.17.
(ii)⇒(iv). By (ii) the representation λG ⊗ λ̄G has almost invariant vectors. But this
representation is unitarily equivalent to a direct sum of |G| copies of λG , since for every
t ∈ G the isometric linear map from ℓ2 (G) to ℓ2 (G) ⊗ ℓ2 (G) determined on canonical
basis vectors by δs 7→ δs ⊗ δst conjugates λG into λG ⊗ λ̄G , and these conjugates are
pairwise orthogonal. We thus obtain (iv) in view of Lemma 3.18.
3.3. ERGODICITY, WEAK MIXING, AND THE MEAN ERGODIC THEOREM
103
(iv)⇒(i). From (iv) we obtain a sequence {fn } of unit vectors in ℓ2 (G) such that
kλG (s)fn − fn k2 → 0. Observe that for any unit vector f ∈ ℓ2 (G) the function |f |2 is an
element of ℓ1 (G) of norm one, and by the Cauchy-Schwarz inequality we have, for every
s ∈ G,
s|f |2 − |f |2 ≤ λG (s)f λG (s)f¯ − f¯ + λG (s)f − f f¯
1
1
1
≤ kf k2 kλG (s)f − f k2 + kλG (s)f − f k2 kf¯k2
= 2kλG (s)f − f k2 .
It follows that the sequence {|fn |2 } is a witness to Reiter’s property, so that G is amenable
by Theorem 3.4.
R EMARK 3.20. A weakly mixing unitary representation π of an amenable G can fail
to have almost invariant vectors. For example, let µ be any atomless Borel probability
measure on T which does not contain 1 in its support. Then it is readily seen that the
representation of Z on L2 (T, µ) which sends n to multiplication by the function z 7→ z n
is weakly mixing but does not have almost invariant vectors. Compare however Theorem 4.9.
3.3. Ergodicity, weak mixing, and the mean ergodic theorem
An immediate consequence of Proposition C.17 and Theorem 1.19 is that, for p.m.p.
actions of an amenable G, both ergodicity and weak mixing can be expressed as asymptotic conditions involving averages along a Følner sequence:
T HEOREM 3.21. Suppose that G is amenable, and let {Fn } be a (left) Følner sequence
for G. Let G y (X, µ) be a p.m.p. action. Then the action is ergodic if and only if
1 X
lim
µ(sA ∩ B) = µ(A)µ(B)
n→∞ |Fn |
s∈F
n
for all measurable A, B ⊆ X, and weakly mixing if and only if
1 X
|µ(sA ∩ B) − µ(A)µ(B)| = 0
lim
n→∞ |Fn |
s∈F
n
for all measurable A, B ⊆ X.
The above characterization of ergodicity can be established in a more direct way by
applying the following mean ergodic theorem to vectors of the form 1A −µ(A)1 in L2 (X).
This mean ergodic theorem refines the abstract mean ergodic theorem (Theorem 1.19) to
a more concrete and useful form which is stronger both in the combinatorial expression
of the mean as an asymptotic averaging over Følner sets and in the application of this
104
3. AMENABILITY
averaging to single vectors in the Hilbert space. One can similarly prove a version for
right Følner sequences, with π(s−1 ) being replaced by π(s).
T HEOREM 3.22. Suppose that G is amenable, and let {Fn } be a (left) Følner sequence
for G. Let π : G → B(H) be a unitary representation. Then
1 X
−1
=0
lim π(s
)ξ
−
P
ξ
n→∞ |Fn |
s∈Fn
for all ξ ∈ H, where P is the orthogonal projection of H onto the subspace of G-invariant
vectors.
P ROOF. As observed in the proof of Theorem 1.19, the subspace of G-invariant vectors is the orthogonal complement of the set of vectors of the form π(t)η − η for some
η ∈ H and t ∈ G. The convergence in the theorem being trivial for G-invariant vectors,
it thus suffices by a linear span and approximation argument to check the convergence on
vectors of the form π(t)η − η, and in this case we indeed have
1 X
1 X
−1
−1
−1
=
π(s
)(π(t)η
−
η)
(π(s
t)
−
π(s
))η
|Fn |
|Fn |
s∈Fn
s∈Fn
X
X
1
−1
−1
=
π(s
)η
−
π(s
)η
|Fn |
−1
−1
s∈t
Fn \Fn
|t−1 Fn ∆Fn |
kηk
|Fn |
→0
s∈Fn \t
Fn
≤
as n → ∞.
3.4. The pointwise ergodic theorem
When applied to the Koopman representation of a p.m.p. action of an amenable group,
the convergence in the mean ergodic theorem (Theorem 3.22) can be upgraded to pointwise a.e. convergence in the more general setting of L1 functions (Theorem 3.27), granted
that we impose a temperedness condition on our Følner sequence (Definition 3.25). In the
classical Birkhoff form of the pointwise ergodic theorem, when the group is Z, the averaging is done over the intervals {0, . . . , n − 1}, which can be seen as the prototype of a
tempered Følner sequence.
L EMMA 3.23. Let F and C be nonempty finite subsets of G, µ a positive measure on
G, and λ > 0, and suppose that µ(F c) ≥ λ|F | for every c ∈ C. Then there is a D ⊆ C
such that µ(F D) ≥ λ max( 21 |F ||D|, 41 |C|).
3.4. THE POINTWISE ERGODIC THEOREM
105
P ROOF. Take a maximal subset D ⊆ C with the property that there exist pairwise
disjoint sets Ad ⊆ F d for d ∈ D such that µ(Ad ) ≥ 21 µ(F d) for every d ∈ D. We then
have
X
X1
λ
µ(F D) ≥
µ(Ad ) ≥
µ(F d) ≥ |F ||D|.
2
2
d∈D
d∈D
Now if |D| ≥ 12 |C| then we are done, and so we may assume that |D| ≤
c ∈ C\D we have µ(F c ∩ F D) ≥ 12 µ(F c) and thus
X
1 X
µ(F c) ≤
µ(F c ∩ F D) ≤ |F |µ(F D)
2
c∈C\D
1
|C|.
2
For
c∈C\D
where the second inequality follows from the fact that each element of G can belong to at
most |F | of the sets F c ∩ F D for c ∈ C\D. Therefore
1 X
λ
λ
µ(F D) ≥
µ(F c) ≥ |C\D| ≥ |C|,
2|F |
2
4
c∈C\D
completing the proof.
L EMMA 3.24. Let F1 , . . . , Fn and C1 , . . . , Cn be finite subsets of G and µ a positive
S
measure
on afinite subset E of G containing nk=1 Fk Ck . Let b, λ > 0 be such that
Sk−1 −1
i=1 Fi Fk ≤ b|Fk | for all k = 2, . . . , n and µ(Fk c) ≥ λ|Fk | for all c ∈ Ck and
k = 1, . . . , n. Then
n
[ 2(b + 2)
µ(E).
Ck ≤
λ
k=1
P ROOF. For k running from n down to 1 we recursively construct sets Dk ⊆ Ck′ such
that µ(Fk Dk ) ≥ λ max( 12 |Fk ||Dk |, 14 |Ck′ |) and F1 D1 , . . . , Fn Dn are pairwise disjoint by
applying Lemma 3.23 at the kth stage to Fk and
S
S
Ck′ := Ck \ ni=k+1 Fk−1 Fi Di = c ∈ Ck : Fk c ∩ ni=k+1 Fi Di = ∅
when these are nonempty (taking Cn′ = Cn in the base case k = n). Then
n
n i−1
n i−1
[
[ [
X
[ −1 −1
′
Ck \Ck = Ck ∩ Fk Fi Di ≤
Fk Fi |Di |
k=1
≤b
i=2 k=1
n
X
i=2
|Fi ||Di | ≤
i=2
n
2b X
λ
i=2
k=1
µ(Fi Di ) ≤
2b
µ(E)
λ
and consequently
[
[
X
n
n
n
n
4X
2b
′
′
Ck ≤ Ck \Ck +
|Ck | ≤ µ(E) +
µ(Fk Dk )
λ
λ
k=1
k=1
k=1
k=1
106
3. AMENABILITY
≤
2(b + 2)
µ(E),
λ
as desired.
∞
D EFINITION 3.25. A sequence
n }n=1
of finite subsets of G is said to be tempered
Sk−1 {F
−1
if there is a b > 0 such that i=1 Fi Fk ≤ b|Fk | for every k ≥ 2.
Let G y (X, µ) be a p.m.p. action and let f ∈ L1 (X). For a nonempty finite set
F ⊆ G we define on X the averaging function
1 X
AF f (x) =
f (sx),
|F | s∈F
and for a tempered sequence {Fn }∞
n=1 in G we define on X the maximal function
M f (x) = sup AFn |f |(x).
n∈N
T HEOREM 3.26. Let {Fn }∞
n=1 be a tempered Følner sequence in G and b > 0 a constant witnessing the temperedness condition. Then, for every p.m.p. action G y (X, µ),
µ{x ∈ X : M f (x) > λ} ≤
2(b + 2)
kf k1
λ
for all f ∈ L1 (X) and λ > 0.
P ROOF. Let ε > 0. Choose an n ∈ N large enough so that the set B := {x ∈ X :
Mn f (x) > λ} has µ-measure at least µ{x ∈ X : M f (x) > λ} − ε, where Mn f (x) =
max1≤k≤n AFk |f |(x). Since G is amenable we can find a nonempty finite set F ⊆ G such
Sn
that the set E =
k=1 Fk F satisfies |E| ≤ (1 + ε)|F |.
For a given x ∈ X we set Ck = {s ∈ F : AFk |f |(sx) > λ} for k = 1, . . . , n and
apply Lemma 3.24 with respect to the measure on E with density function s 7→ |f (sx)|
to obtain
n
X
[ 2(b + 2) X
|f (sx)|.
1B (sx) = Ck ≤
λ
s∈E
s∈F
k=1
Integrating over x and using the G-invariance of µ,
Z X
1
µ(B) =
1B (sx) dµ(x)
|F | X s∈F
Z X
2(b + 2)
≤
|f (sx)| dµ(x)
λ|F |
X s∈E
2(b + 2)
(1 + ε)kf k1 .
λ
Since ε was arbitrary this yields the result.
≤
3.4. THE POINTWISE ERGODIC THEOREM
107
Let G y (X, µ) be a p.m.p. action. The set V of G-invariant functions in L1 (X)
forms a closed subspace,R and there is aR norm-one idempotent positive linear map E :
L1 (X) → V satisfying X E(f ) dµ = X f dµ for all f ∈ L1 (X) (the conditional expectation onto V ), which can be defined as follows. First define E on L2 (X) as the
orthogonal projection onto the closed subspace V ′ of G-invariant functions in L2 (X).
Given an f ∈ L2 (X) we define the function h = E(f )/|E(f )|, interpreted to be zero at
points where E(f ) is zero, and note that h ∈ V ′ . Then hf − E(f ), hi = 0, and so
Z
f h dµ ≤ kf k1 .
kE(f )k1 =
X
Thus E extends uniquely to a norm-one linear map L1 (X) → V , which can be checked
to have the desired properties.
T HEOREM 3.27. Let G y (X, µ) be a p.m.p. action. Let {Fn }∞
n=1 be a tempered
1
Følner sequence in G and f ∈ L (X). Then
lim AFn f (x) = Ef (x)
n→∞
µ-a.e.
where E is the conditional expectation onto the subspace of G-invariant functions in
L1 (X).
P ROOF. We may assume our spaces to be over the real numbers by decomposing into
real and imaginary parts. Since the theorem obviously holds for all f in the image W of
E, we need only establish the conclusion for the closed subspace V consisting of those
f ∈ L1 (X) such that E(f ) = 0.
Observe that if g ∈ L∞ (X) and s ∈ G then for µ-a.e. x ∈ X we have
|sFn ∆Fn |
lim sup AFn (g − sg)(x) ≤ lim
kgk∞ = 0.
n→∞
|Fn |
n→∞
It follows by the triangle inequality that limn→∞ AFn f (x) = 0 µ-a.e. for all f in the linear
subspace Z spanned by functions of the form g − sg for such g and s. Note that Z ⊆ V
since E(sf ) = sE(f ) = E(f ) for all f ∈ L1 (X) and s ∈ G.
We next argue that Z is dense in V . It suffices to prove that Z + W is dense in L1 (X),
since the latter decomposes topologically as V ⊕W and Z ⊆ V . Suppose then that we are
given an h ∈ (Z + W )⊥ ⊆ L1 (X)∗ = L∞ (X) and let us demonstrate that h = 0, which
will yield the density of Z + W by the Hahn-Banach theorem. Using the G-invariance of
µ we have, for all g ∈ L1 (X) and s ∈ G,
Z
Z
−1
(h − sh)g dµ
h(g − s g) dµ =
0=
X
X
R
so that sh = h and hence h ∈ W . But since h ∈ W ⊥ this means that X h2 dµ = 0 and
thus h = 0, as desired.
108
3. AMENABILITY
Now let f be an arbitrary function in V . Given an ε > 0, we can write f = f1 + f2
where f1 ∈ Z and kf2 k1 < ε. Taking a constant b > 0 that witnessses the temperedness
condition and applying Theorem 3.26 to f2 yields
√ 2(b + 2)
√
µ x ∈ X : M f2 (x) > ε ≤ √
kf2 k1 ≤ 2(b + 2) ε.
ε
√
Then for all x outside a set of measure at most 2(b + 2) ε we have
√
lim sup |AFn f (x)| ≤ lim |AFn f1 (x)| + M f2 (x) ≤ ε.
n→∞
n→∞
Since ε was arbitrary we obtain the statement of the theorem.
3.5. Quasitilings and the subadditivity theorem
The quasitiling theorem, Theorem 3.35, tells us that an approximately invariant finite
subset A of a group G can approximately be tiled by translates of finitely many nested
nonempty sets T1 ⊆ · · · ⊆ Tn each of which except the first is sufficiently approximately
invariant with respect to the previous one. The number n of these sets grows in an unbounded way as the tolerance of the approximation tends to zero, and does not depend
on the group. The proof of the theorem begins by first covering A with translates of Tn
and then extracting a subcollection of these translates which is almost disjoint and covers
as much of A as possible, which however will only be a proportionally small amount.
Now we repeat the procedure by replacing A with the complement of the union of these
translates and using the next-to-last set, and continue recursively until the tiles have been
exhausted, at which point A will be almost entirely covered.
What is curious about this argument, which tiles more and more of A but only in
small increments with sets of widely varying size, is that it does not reflect our experience
with concrete examples. Indeed for many amenable groups, such as Abelian groups,
the quasitiling can be accomplished by translates of a single set and in a global way
across the whole group, and in fact there is no example which is known not to satisfy this
“monotilability” property [145].
The quasitiling theorem does not require per se that G be amenable, but it is precisely
in the amenable case that the hypotheses will be nonvacuous across all ε in the given
range. Thus in this section we will not assume the group G to be amenable until we
reach the subadditivity theorem, Theorem 3.37, which we include here as an important
application of quasitiling.
The following definitions and two lemmas will be applied in this section to finite
subsets of G, in which case counting measure is used. In the next section we will apply
them to actions in order to establish the Rokhlin lemma.
3.5. QUASITILINGS AND THE SUBADDITIVITY THEOREM
109
D EFINITION 3.28. Let (X, µ) be a finite measure space. Let λ, ε ≥ 0. We say that a
collection {Ai }i∈I of measurable subsets of X
P
(i) is a λ-even covering of X if there exists a positive integer M such that i∈I 1Ai ≤
P
M and i∈I µ(Ai ) ≥ λM µ(X), in which case M is called a multiplicity of the
λ-even covering,
S
(ii) λ-covers X if µ( i∈I Ai ) ≥ λµ(X),
bi ⊆ Ai such that µ(A
bi ) ≥
(iii) is ε-disjoint if there exist pairwise disjoint sets A
(1 − ε)µ(Ai ) for all i ∈ I.
L EMMA 3.29. Let (X, µ) be a finite measure space. Let λ ∈ (0, 1) and let {Ai }i∈I be
a λ-even covering of X. Then for every nonnull B ⊆ X there exists an i ∈ I such that
µ(B)
µ(Ai ∩ B)
≤
.
µ(Ai )
λµ(X)
P ROOF. If for some B ⊆ X we had
µ(Ai ∩ B) >
µ(B)
µ(Ai )
λµ(X)
for every i ∈ I, then taking a multiplicity M for the λ-even covering and summing over i
would yield
X
µ(B) X
µ(Ai ) ≥ µ(B)M
µ(Ai ∩ B) >
λµ(X)
i∈I
i∈I
X
Z
≥
1B (x)
1Ai (x) dµ(x)
X
=
Z X
X
=
X
i∈I
a contradiction.
i∈I
i∈I
1Ai ∩B (x) dµ(x) =
XZ
i∈I
X
1Ai ∩B (x) dµ(x)
µ(Ai ∩ B),
L EMMA 3.30. Let (X, µ) be a finite measure space. Let 0 ≤ ε < 1 and 0 < λ ≤ 1
and let {Ai }i∈I be a λ-even covering of X by nonnull sets. Then there is an ε-disjoint
subcollection of {Ai }i∈I which ελ-covers X.
P ROOF. By Zorn’s lemma there exists a maximal ε-disjoint subcollection {Ai }i∈J of
{Ai }i∈I . If this does not ελ-cover X then by Lemma 3.29 there is an i0 ∈ I such that
S
S
µ i∈J Ai
µ Ai0 ∩ i∈J Ai
≤
<ε
µ(Ai0 )
λµ(X)
110
3. AMENABILITY
so that by adding Ai0 to the collection {Ai }i∈J we again have an ε-disjoint collection,
contradicting maximality.
We next formalize the notion of approximate invariance of one finite subset of the
group G with respect to another.
D EFINITION 3.31. Let F and A be nonmepty finite subsets of G. We say that A is
(F, δ)-invariant if |{s ∈ A : F s ⊆ A}| ≥ (1 − δ)|A|.
L EMMA 3.32. Let δ > 0 and let F, A ⊆ G be nonempty finite sets such that A is
(F, δ)-invariant. Then the collection {F s : s ∈ A and F s ⊆ A} is a (1 − δ)-even
covering of A with multiplicity |F |.
P ROOF. The set I of all s ∈ A such that F s ⊆ A has cardinality at least (1 − δ)|A| by
P
(F, δ)-invariance, so that s∈I |F s| = |I||F | ≥ (1 − δ)|A||F |. On the other hand, each
P
element of G belongs to at most |F | translates of F , and so s∈I 1F s ≤ |F |.
D EFINITION 3.33. Let F and A be finite subsets of the group G. We define the F boundary of A by
∂F A = {s ∈ G : F s ∩ A 6= ∅ and F s ∩ Ac 6= ∅},
T
S
which can also be written as s∈F s−1 A\ s∈F s−1 A.
Although we have defined A to be (F, δ)-invariant if |{s ∈ A : F s ⊆ A}| ≥
(1 − δ)|A|, we could also express approximate invariance by requiring that |∂F A| be
proportionally small with respect to |A|. Such a condition will be used in Theorem 3.35
below and can be substituted into the Følner characterization of amenability. Indeed asT
suming e ∈ F we have {s ∈ A : F s ⊆ A} = s∈F s−1 A and if the cardinality of this set
is less than (1 − δ)|A| then
[ −1 \ −1 |∂F A| = s A − s A > |A| − (1 − δ)|A| = δ|A|.
s∈F
s∈F
Conversely, if A is (F F −1 , δ/(2|F |2 ))-invariant then |tA∆A| ≤ δ|A|/|F |2 for all t ∈
F F −1 and so
[
X
−1
−1 ts−1 A∆A ≤ δ|A|.
s A∆t A ≤
|∂F A| = s,t∈F
s,t∈F
D EFINITION 3.34. Let A be a finite subset of G and ε > 0. A finite collection
{T1 , . . . , Tn } of finite subsets of G is said to ε-quasitile A if there exist C1 , . . . , Cn ⊆ G
S
S
such that ni=1 Ti Ci ⊆ A and the collection of right translates ni=1 {Ti c : c ∈ Ci } is
ε-disjoint and (1 − ε)-covers A.
3.5. QUASITILINGS AND THE SUBADDITIVITY THEOREM
111
T HEOREM 3.35. Let 0 < ε < 12 and let n be a positive integer such that (1 − ε/2)n <
ε. Then whenever e ∈ T1 ⊆ T2 ⊆ · · · ⊆ Tn are finite subsets of G with |∂Tk−1 Tk | ≤
(ε/8)|Tk | for k = 2, . . . , n, every (Tn , ε/4)-invariant finite subset of G is ε-quasitiled by
{T1 , . . . , Tn }.
P ROOF. Suppose we are given finite subsets e ∈ T1 ⊆ T2 ⊆ · · · ⊆ Tn of G with
|∂Tk−1 Tk | ≤ (ε/8)|Tk | for k = 2, . . . , n. Let A be a nonempty (Tn , ε/4)-invariant finite
subset of G. We will recursively construct Cn , Cn−1 , . . . , C1 ⊆ G such that, for each
S
S
k = 1, . . . , n, ni=k Ti Ci ⊆ A and the collection of translates ni=k {Ti c : c ∈ Ci } is
ε-disjoint and λ-covers A where λ is the minimum of 1 − ε and 1 − (1 − ε/2)n−k+1 . It
will then follow by our assumption on n that {T1 , . . . Tn } ε-quasitiles A.
For the first step, we note that by Lemma 3.32 the collection of all right translates of
Tn that lie in A is a (1 − ε)-even covering of A. Then by Lemma 3.30 we can find an
ε-disjoint subcollection {Tn c : c ∈ Cn } of these translates which (ε/2)-covers A.
Suppose now that for some k ∈ {1, . . . , n−1} we have constructed Cn , Cn−1 , . . . , Ck+1 ⊆
S
G with the desired property. Set Ak = A\ ni=k+1 Ti Ci . If |Ak | < ε|A| then we can finish
the entire construction by taking each of Ck , Ck−1 , . . . , C1 to be the empty set. So we may
assume that |Ak | ≥ ε|A|. Let us show that Ak is (Tk , 12 )-invariant. For i = k +1, . . . , n we
have |∂Tk (Ti c)| ≤ |∂Ti−1 (Ti c)| = |∂Ti−1 T i | ≤ (ε/8)|Ti| for all c ∈ Ci . Since the collection
Sn
1
−1
Sn
i=k+1 {Ti c : c ∈ Ci } is 2 -disjoint and
i=k+1 Ti Ci ≤ |A| ≤ ε |Ak |, we thus obtain
n
[
n
ε
n [
ε X
ε [
1
|Ti ||Ci | ≤ Ti Ci ≤ |A| ≤ |Ak |.
∂Tk (Ti c) ≤
8 i=k+1
4 i=k+1
4
4
i=k+1 c∈C
i
Writing J for the set of all s ∈ A such that Tk s ⊆ A, we have |J| ≥ (1 − ε/4)|A| by the
(Tk , ε/4)-invariance of A and consequently
n [
[
{s ∈ Ak : Tk s ⊆ Ak } = J\
T i Ci ∪
∂Tk (Ti c) i=k+1
c∈Ci
[
[
n
n [
Ti Ci − ≥ |J| − ∂Tk (Ti c)
≥ 1−
i=k+1
ε
i=k+1 c∈Ci
1
|A| − (|A| − |Ak |) − |Ak |
4
4
1
≥ |Ak |,
2
that is, Ak is (Tk , 21 )-invariant. Thus by Lemma 3.32 the right translates of Tk that lie in Ak
form a 21 -even covering of Ak , and so by Lemma 3.30 there is an ε-disjoint subcollection
S
{Tk c : c ∈ Ck } of these translates which (ε/2)-covers Ak . It follows that ni=k {Ti c : c ∈
112
3. AMENABILITY
Ci } is ε-disjoint and (1−(1−ε/2)n−k+1 )-covers A, completing the recursive construction
and the proof of the theorem.
D EFINITION 3.36. Let f be a real-valued function on the set of all finite subsets of
G. We say that f (A) converges to a limit L as A becomes more and more invariant if for
every ε > 0 there are a finite set F ⊆ G and a δ > 0 such that |f (A) − L| < ε for every
nonempty (F, δ)-invariant finite set A ⊆ G.
T HEOREM 3.37. Suppose that G is amenable. Let ϕ be a [0, ∞)-valued function on
the set of all finite subsets of G such that
(i) ϕ(As) = ϕ(A) for all finite A ⊆ G and s ∈ G,
(ii) ϕ(A ∪ B) ≤ ϕ(A) + ϕ(B) for all finite A, B ⊆ G.
Then ϕ(A)/|A| converges to a limit as A becomes more and more invariant.
P ROOF. For a finite F ⊆ G and δ > 0 write I(F, δ) for the collection of nonempty
(F, δ)-invariant finite subsets of G. The pairs (F, δ) form a net Λ where (F ′ , δ ′ ) ≻ (F, δ)
means that F ′ ⊇ F and δ ′ ≤ δ. Set
ϕ(A)
a0 = lim
inf
(F,δ)∈Λ A∈I(F,δ) |A|
Note that this limit is finite since ϕ(A) ≤ |A|ϕ({e}) for every nonempty finite set A ⊆ G.
Let η > 0. To establish the theorem it is enough to show the existence of a pair
(F, δ) ∈ Λ such that ϕ(A)/|A| < a0 + η for all A ∈ I(F, δ). Let 0 < ε < 12 , and
take an N ∈ N as given by Theorem 3.35 relative to ε. Then there exist finite subsets
e ∈ T1 ⊆ T2 ⊆ · · · ⊆ TN of G that ε-quasitile every (TN , ε/4)-invariant finite subset of
G and satisfy
η
ϕ(Ti )
(19)
≤ a0 +
|Ti |
2
for every i = 1, . . . , N .
Now suppose that we are given an A ∈ I(TN , ε/4). Then there is an ε-disjoint colS
lection {Ti cij }i,j of translates of the Ti that (1 − ε)-covers A with i,j Ti cij ⊆ A. By the
ε-disjointness of the translates Ti cij there exist Td
i cij ⊆ Ti cij which are pairwise disjoint
d
and satisfy |Ti cij |/|Ti cij | ≥ 1 − ε, so that
X
X
X Td
i cij
Td
|A| ≥
(20)
|Ti cij | ≥ (1 − ε)
|Ti |.
i cij =
|T
c
|
i
ij
i,j
i,j
i,j
Using (19) and (20) and the properties of ϕ, we have
[
[
ϕ(A) ≤ ϕ
Ti cij + ϕ A\ Ti cij
i,j
i,j
3.6. THE ROKHLIN LEMMA
X
113
η
+ ε|A|ϕ({e})
2
i,j
η
1 a0 +
+ εϕ({e}) .
≤ |A|
1−ε
2
≤
|Ti |
a0 +
Thus by taking ε small enough we will obtain ϕ(A)/|A| < a0 + η, completing the proof.
3.6. The Rokhlin lemma
Here we establish the Rokhlin lemma (Theorem 3.43) of Ornstein and Weiss for free
p.m.p. actions and use it to show that any two free p.m.p. actions of a given amenable G
are approximately conjugate (Theorem 3.44). The latter fact will also be used to derive
a version of the Rokhlin lemma in Theorem 3.45 that replaces quasitowers with genuine
towers. These results require that the probability space (X, µ) be atomless, and so we will
make this a blanket assumption throughout the section.
The Rokhlin lemma is a nondiscrete version of the quasitiling theorem of the previous
section, with tiles within the group being replaced by towers within the probability space,
and its proof follows the same basic principle. For the quasitiling theorem we implicitly
used the fact the left translation action of a group on itself is free, along with the discreteness of the group, in order to produce tiles as translates of an initial set. In the present
context we impose freeness as an assumption on our action, while the atomlessness of
the space compels us to apply a maximality argument in order to find sufficiently many
towers. As for the quasitiling theorem, Theorem 3.43 does not require G to be amenable,
although it is only in that case that the hypotheses are nonvacuous across all ε in the given
range. We will thus not assume that the group G is amenable unless explicitly stated.
For a p.m.p. action G y (X, µ) we make the following definitions. The first is the
analogue of Definition 3.31 for p.m.p. actions.
D EFINITION 3.38. Let F be a finite susbet of G and δ > 0. A measurable set A ⊆ X
T
is (F, δ)-invariant if µ(A ∩ t∈F t−1 A) ≥ (1 − δ)µ(A).
D EFINITION 3.39. Given a finite set F ⊆ G, we say that a set V ⊆ X is the base of
an F -tower if the map F × V → X sending (s, x) to sx is injective.
Using this last piece of terminology we can restate Proposition 1.4 as follows.
L EMMA 3.40. Let G y (X, µ) be a free p.m.p. action. Let F be a finite subset of G
and A a nonnull subset of X. Then there is a nonnull subset of A which is the base of an
F -tower.
114
3. AMENABILITY
L EMMA 3.41. Let G y (X, µ) be a free p.m.p. action. Let F be a finite subset of G
containing e and let A be a nonnull subset of X. Then there is a partition {Ai }i∈I of A
modulo null sets such that each Ai is nonnull and is the base of an F -tower. Moreover, if
for a given δ > 0 the set A is (F, δ)-invariant then the collection of sets F (Ai ∩ AF ) ⊆ A
T
for i ∈ I is a (1 − δ)-even covering of A with multiplicity |F |, where AF = t∈F t−1 A .
P ROOF. By Zorn’s lemma there is a maximal family {Ai }i∈I of pairwise disjoint nonnull subsets of A which are bases of F -towers, and it follows by Lemma 3.40 and maxiS
mality that i∈I Ai is equal to A modulo a null set, yielding the first statement.
Now let δ > 0 and suppose that A is (F, δ)-invariant. Then
X
X
µ(F (Ai ∩ AF )) =
|F |µ(Ai ∩ AF ) = |F |µ(AF ) ≥ (1 − δ)|F |µ(A).
i∈I
i∈I
On the other hand, each x ∈ A can belong to at most |F | sets of the form F Ai since the Ai
P
are pairwise disjoint, and so i∈I 1F (Ai ∩AF ) ≤ |F |, yielding the second statement.
D EFINITION 3.42. Let G y (X, µ) be a p.m.p. action on an atomless space. For a
finite set F ⊆ G and a measurable set V ⊆ X we say that F × V maps to an ε-quasitower
if there exists a measurable subset D ⊆ F × V such that the restriction to D of the map
F × V → X sending (s, x) to sx is injective and for almost every x ∈ V the cardinality
of {s ∈ F : (s, x) ∈ D} is at least (1 − ε)|F |.
T HEOREM 3.43. Let G y (X, µ) be a free p.m.p. action with µ atomless. Let 0 < ε ≤
and let n be an integer such that (1 − ε/16)n < ε/2. Let e ∈ T1 ⊆ T2 ⊆ · · · ⊆ Tn be
finite subsets of G with |∂Tk−1 Tk | ≤ 6ε |Tk | for k = 2, . . . , n. Then there exist measurable
sets V1 , . . . , Vn ⊆ X such that
(i) each Tk × Vk maps to an ε-quasitower,
(ii) Tk Vk ∩ Tk′ Vk′ = ∅ for k 6= k ′ ,
S
(iii) µ nk=1 Tk Vk ≥ 1 − ε,
and each of the quantities µ(V1 ), . . . , µ(Vn ) does not depend on the action.
1
2
P ROOF. Denote by m the smallest integer satisfying (1 − ε/16)m < ε/2. We will
recursively construct sets Vk for k running from n down to n − m + 1 such that, for each
k,
(i) each Tk × Vk maps to an (ε/2)-quasitower,
(ii) Tk Vk ∩ Tk′ Vk′ = ∅ for k ′ 6= k,
ε
(1 − ε/16)n−k .
(iii) µ(Vk ) = |Tk |−1 16
We will then have
[
n
n
X
ε
1−
|Ti |µ(Vi )
µ
Ti Vi ≥
2
i=n−m+1
i=n−m+1
3.6. THE ROKHLIN LEMMA
115
ε ε ε n−i
1−
2 16
16
i=n−m+1
ε m ε 1− 1−
= 1−
2
16
ε 2
≥ 1−
≥ 1 − ε,
2
=
n
X
1−
and so these sets together with Vk = ∅ for 1 ≤ k ≤ n − m will satisfy the requirements
of the theorem statement.
For the first step, apply Lemma 3.41 to obtain a partition P of X modulo null sets such
that each element of P has positive measure and is the base of a Tn -tower. Then {Tn W :
W ∈ P} is a 23 -even covering of X and so by applying Lemma 3.30 we can find, after
discarding cofinitely many sets whose union has small measure, sets Wn,1 , . . . , Wn,ln ∈ P
Sn
Tn Wn,j is an
such that the family {Tn Wn,j : j = 1, . . . , ln } is (ε/4)-disjoint and lj=1
(ε/8)-cover of X. To complete the first step, jump to the paragraph after the following
one and apply the argument there with k = n.
Suppose now that for some k ∈ {n − m + 1, . . . , n − 1} we have constructed sets Vi
S
for i = k + 1, . . . , n with the desired properties. Set Xk = X\ ni=k+1 Ti Vi . Then
µ
[
n
i=k+1
Ti Vi
n
X
ε n−i
ε n−k
ε
1−
=1− 1−
,
≤
|Ti |µ(Vi ) =
16
16
16
i=k+1
i=k+1
n
X
and hence
µ(Xk ) = 1 − µ
[
n
i=k+1
Ti Vi
ε n−k
.
≥ 1−
16
m−1
Since (1 − ε/16)
≥ ε/2, we have
[
n
ε m−1
ε
ε n−(n−m+1)
µ
=1− 1−
≤1−
Ti Vi ≤ 1 − 1 −
16
16
2
i=k+1
and
ε n−(n−m+1) ε m−1 ε
µ(Xk ) ≥ 1 −
= 1−
≥ .
16
16
2
Let us show that Xk is (Tk , 12 )-invariant. In analogy with Definition 3.33, for sets F ⊆ G
T
S
and A ⊆ X we write ∂F A for the set t∈F t−1 A\ t∈F t−1 A. For each i = k + 1, . . . , n
we have
[
[
\
\
−1
−1
−1
−1
t Ti Vi \
t Ti Vi ⊆
t Ti \
t Ti Vi
t∈Tk
t∈Tk
t∈Tk
t∈Tk
116
3. AMENABILITY
and hence µ(∂Tk (Ti Vi )) ≤ |∂Tk Ti |µ(Vi ) ≤ 6ε |Ti |µ(Vi ). By conditions (i) and (ii) we thus
obtain
[
[
n
n
n
ε
ε X
−1
Ti Vi
|Ti |µ(Vi ) ≤ (1 − ε/2) µ
µ
∂Tk (Ti Vi ) ≤
6 i=k+1
6
i=k+1
i=k+1
≤
and consequently
\
n
[
−1
µ
t Xk = µ X\
t∈Tk
i=k+1
≥ µ(Xk ) − µ
ε
1
≤ µ(Xk )
6
3
Ti Vi ∪ ∂Tk (Ti Vi )
[
n
∂Tk (Ti Vi )
i=k+1
2
≥ µ(Xk ),
3
(Tk , 13 )-invariant.
that is, Xk is
It follows by Lemma 3.41 that there is a partition Q of Xk
modulo null sets such that each
of Q has positive
measure and is the base of ′ a
element
T
′
−1
Tk -tower and, writing Q = P ∩ t∈Tk t Xk : P ∈ Q , the family {Tk W : W ∈ Q }
is a 23 -even covering of Xk . Then by Lemma 3.30 we can find, after discarding cofinitely
many sets whose union has small measure, sets Wk,1 , . . . , Wk,lk ∈ Q′ such that the family
Sk
{Tk Wk,j : j = 1, . . . , lk } is (ε/4)-disjoint and lj=1
Tk Wk,j is a (ε/8)-covering of Xk .
S lk
We will show that the set Wk := j=1 Wk,j can be shrunk slightly to obtain the
desired Vk . By (ε/4)-disjointness we can find for j = 1, . . . , lk pairwise disjoint sets
′
\
T\
k Wk,j ⊆ Tk Wk,j satisfying µ(Tk Wk,j ) ≥ (1 − ε/4)µ(Tk Wk,j ). Define Wk,j to be the set
of all x ∈ Wk,j such that |{t ∈ Tk : tx ∈ T\
k Wk,j }| ≥ (1 − ε/2)|Tk |. Then the set of all
′
\
points in Tk Wk,j \Tk Wk,j of the form tx for x ∈ Wk,j \Wk,j
and t ∈ Tk has measure at
′
), and so
least 2ε |Tk |µ(Wk,j \Wk,j
ε
′
|Tk |µ(Wk,j \Wk,j
) ≤ µ(Tk Wk,j \T\
k Wk,j )
2
ε
ε
≤ µ(Tk Wk,j ) = |Tk |µ(Wk,j ).
4
4
S
k
′
Setting Wk′ = lj=1
Wk,j
it follows that
|Tk |µ(Wk′ )
≥ |Tk |
lk X
j=1
1
µ(Wk,j )
1−
2
1
ε
ε
ε n−k
≥ µ(Tk Wk ) ≥ µ(Xk ) ≥
1−
,
2
16
16
16
ε
(1 −
and thus using Theorem A.20 we can find a set Vk ⊆ Wk′ such that µ(Vk ) = |Tk |−1 16
n−k
ε/16) . Note finally that the set of all (t, x) ∈ Tk × Vk such that x ∈ Wk,j and
3.6. THE ROKHLIN LEMMA
117
tx ∈ T\
k Wk,j for some 1 ≤ j ≤ lk witnesses the fact that Tk × Vk maps to an (ε/2)quasitower, completing the recursive construction.
Write Aut(X, µ) for the group of all measure-preserving transformations of X modulo the relation of almost eveywhere agreement. In accord with the usual conventions we
typically think of and write the elements of Aut(X, µ) as if they were genuine automorphisms. On Aut(X, µ) we define the metric
dµ (R, S) = µ({x ∈ X : Rx 6= Sx}),
The topology induced by this metric is much stronger than the weak topology that appears
in Section ?? and is not Polish. The following striking application of Theorem 3.43 asserts
that any two free p.m.p. actions of a given amenable G are approximately conjugate with
respect to d to within any tolerance. It is a hallmark of amenability that one can make
such a general perturbative statement with respect to such a strong notion of proximity.
T HEOREM 3.44. Suppose that G is amenable. Let R and S be free p.m.p. actions of
G on (X, µ) with µ atomless. Let δ > 0 and let F be a finite subset of G. Then there is a
U ∈ Aut(X, µ) such that dµ (Rs , U −1 Ss U ) < δ for all s ∈ F .
P ROOF. The idea is immediate from the Rokhlin lemma (Theorem 3.43): we take
Rokhlin tower (or more precisely quasitower) decompositions for the two actions and
then match these up to produce an approximate conjugacy. The matching can only be
done on most of X because of the possibility of small overlaps in the levels of the towers.
The crucial point is the universality of the Rokhlin tower decomposition over all free
p.m.p. actions of G: at a given degree of tolerance in the modeling of the action, both the
measure of the tower bases and the Følner sets generating the towers are predetermined.
We now describe the procedure in more detail. We may assume that e ∈ F . Take an
ε ∈ (0, 21 ] such that 2(|F | + 1)ε < δ. By Theorem 3.43, we can find (F, ε)-invariant finite
sets T1 , . . . , Tn ⊆ G for which there exist measurable sets V1 , . . . , Vn ⊆ X such that, with
respect to the action R,
(i) each Tk × Vk maps to an ε-quasitower,
(ii) Tk Vk ∩ Tk′ Vk′ = ∅ for k 6= k ′ ,
S
(iii) µ nk=1 Tk Vk ≥ 1 − ε,
and measurable sets V1′ , . . . , Vn′ ⊆ X, with the same properties with respect to the action
S, and µ(Vk′ ) = µ(Vk ) for k = 1, . . . , n. For each k choose a measure isomorphism
Uk : Vk → Vk′ and take sets Dk ⊆ Tk ×Vk and Dk′ ⊆ Tk ×Vk′ witnessing the definition of εquasitower for actions R and S, respectively. Then we can define a measure isomorphism
U : X → X by first setting U Rt x = St Uk x for every (t, x) ∈ Dk with (t, Uk x) ∈ Dk′ and
every k = 1, . . . , n and then extending arbitrarily on the remainder of X.
118
3. AMENABILITY
For a given k, as Tk is (F, ε)-invariant we have |TkF | ≥ (1 − ε)|Tk | where TkF = {t ∈
Tk : F t ⊆ Tk }. Define Ek to be the set of all Rt x such that t ∈ TkF and (st, x) ∈ Dk and
(st, Uk x) ∈ Dk′ for all s ∈ F . With our action notation indicating the application of R to
Vk and S to Vk′ , we have
µ(Ek ) ≥ µ(Tk Vk ) − µ((Tk \TkF )Vk )
X
−
µ({Rt x : (t, x) ∈ TkF × Vk and (st, x) ∈
/ Dk })
s∈F
−
X
s∈F
/ Dk′ })
µ({Rt x : (t, x) ∈ TkF × Vk and (st, Uk x) ∈
≥ µ(Tk Vk ) − ε|Tk |µ(Vk ) − |F |ε|Tk |µ(Vk ) − |F |ε|Tk |µ(Vk )
(2|F | + 1)ε
µ(Tk Vk )
1−ε
1 − 2(|F | + 1)ε
=
µ(Tk Vk )
1−ε
≥ µ(Tk Vk ) −
and so
µ
[
n
k=1
Ek
≥µ
[
n
k=1
≥ (1 − ε)
Tk Vk
1 − 2(|F | + 1)ε
1−ε
1 − 2(|F | + 1)ε
> 1 − δ.
1−ε
Now suppose we are given a y ∈ Ek for some k. Write it as Rt x where t ∈ TkF and
(st, x) ∈ Dk and (st, Uk x) ∈ Dk′ for all s ∈ F . Then, since e ∈ F , for each s ∈ F we
have
U Rs y = U Rst x = Sst Uk x = Ss U Rt x = Ss U y.
Thus dµ (Rs , U −1 Ss U ) < δ for all s ∈ F , as desired.
Using Theorem 3.44 we can now formulate a version of Theorem 3.43 in which the
quasitowers are replaced by genuine towers, i.e., the levels are all disjoint. The trade-off
is a loss of flexibility in the choice of the sets T1 , . . . , Tn .
T HEOREM 3.45. Suppose that G is amenable. Let F be a finite subset of G and let
ε > 0. Then there exist (F, ε)-invariant finite sets T1 , . . . , Tn ⊆ G and κ1 , . . . , κn > 0
such that for every free p.m.p. action G y (X, µ) with µ atomless there exist measurable
sets V1 , . . . , Vn ⊆ X for which
(i) sVk ∩ tVk′ = ∅ for all k, k ′ = 1, . . . , n and all s ∈ Tk and t ∈ Tk′ ,
S
(ii) µ nk=1 Tk Vk > 1 − ε,
(iii) µ(Vk ) = κk for every k = 1, . . . , n.
3.7. THE CONNES-FELDMAN-WEISS THEOREM
119
P ROOF. Fix a free p.m.p. action G y (X, µ) with µ atomless. In view of the Følner
characterization of amenability (Theorem 3.4(i)⇔(v)), we can apply Theorem 3.43 to
obtain nonempty (F, ε)-invariant finite subsets T1 ⊆ T2 ⊆ · · · ⊆ Tn of G, measurable
sets V1 , . . . , Vn ⊆ X, and κ1 , . . . , κn > 0 such that each
Tk × Vk maps to an (ε/2)S
quasitower, Tk Vk ∩ Tk′ Vk′ = ∅ for k 6= k ′ , µ nk=1 Tk Vk ≥ 1 − ε/2, and µ(Vk ) = κk for
every k = 1, . . . , n.
Let k ∈ {1, . . . , n}. By the definition of (ε/2)-quasitower (Definition 3.42), there is a
measurable Dk ⊆ Tk ×Vk such that the restriction to Dk of the map Tk ×Vk → X sending
(s, x) to sx is injective and for almost every x ∈ Vk the cardinality of {s ∈ F : (s, x) ∈
D} is at least (1 − ε/2)|Tk |. Then we can partition Vk into nonnull sets Vk,1 , . . . , Vk,jk so
that for each i = 1, . . . , jk there is a Tk,i ⊆ Tk such that {s ∈ Tk : (s, x) ∈ D} = Tk,i for
all x ∈ Vk,i .
Collecting together all of these sets Tk,i and Vk,i over all k = 1, . . . , n and putting
κk,i = µ(Vk,i ) will give us what the theorem demands, but only for our fixed action.
However, the desired quantification over all free p.m.p. actions of G on atomless spaces
is now made possible by Theorem 3.44.
3.7. The Connes-Feldman-Weiss theorem
In this and the following section we will use the language of Borel sets and maps
instead of the more general qualifier “measurable” in order to stress the decisive role of
standard Borelness to the subject of p.m.p. equivalence relations (Definition 3.48), which
relies heavily on descriptive set theory through results like Theorem 3.47, as well as to
conform with convention in the subject and to economize on syllables.
To set up the terms of the Connes-Feldman-Weiss theorem (Theorem 3.68), we need
to adapt the notion of amenability to p.m.p. equivalence relations (Definition 3.53). The
Connes-Feldman-Weiss theorem tells us that amenable equivalence relations are hyperfinite in the sense of being a.e. equal to the union of an increasing sequence of finite
Borel subequivalence relations. This will help us establish the Ornstein-Weiss theorem in
Section 3.8.
3.7.1. P.m.p. equivalence relations. Let R be a Borel equivalence relation on X,
i.e., an equivalence relation on X which is Borel as a subset of X × X. We will only be
interested in such relations which are countable in the sense that every equivalence classes
is countable, and while we won’t make this a blanket assumption it will be incorported
below into the definition of p.m.p. equivalence relation for terminological economy.
D EFINITION 3.46. A partial transformation is a Borel isomorphism T from one Borel
subset of X to another, in which case we write dom T and im T for the domain and image.
The graph of a partial transformation is the subset {(x, T x) : c ∈ dom T } of X × X,
120
3. AMENABILITY
which we write as gr T . We write JRK for the set of all partial transformations whose
graphs are contained in R.
The graph of a partial transformation T is a Borel subset of X × X. To see this, we
may assume that X is a Polish space and take a countable basis {Un } and then observe
that
∞
\
gr T =
((X\Un ) × X) ∪ (Un × T (Un ∩ dom T )).
n=1
By a graph in a product X × Y we mean a set Q ⊆ X × Y such that if (x, y) ∈ Q
and (x, y ′ ) ∈ Q then y = y ′ (the projection of Q onto X need not be all of X). For any
set K ⊆ X × Y and all x ∈ X and y ∈ Y we define
Kx = K ∩ ({x} × Y ),
K y = K ∩ (X × {y}).
The cornerstone of the theory of countable Borel and p.m.p. equivalence relations is
the following Lusin-Novikov uniformization theorem. It allows us to take properties of
a relation R as expressed in terms of its Borel subsets and translate them into statements
about partial transformations in JRK, as Lemmas 3.55 and 3.65 illustrate. For a proof see
Section 18.C of [82].
T HEOREM 3.47 (Lusin-Novikov uniformization). Let X and Y be standard Borel
spaces and let K be a Borel subset of X × Y such that every section Kx is countable.
Then there is a Borel graph in X × Y with the same projection onto X as K, and this
projection is a Borel subset of X. Moreover, K is a countable union of Borel graphs.
Note that if K is a Borel subset of X × Y which is the graph of a map T defined on a
subset of X, then T is Borel, for if A is a Borel subset of Y then T −1 (A) is equal to the
projection of K ∩ (X × A) onto the first coordinate in X × Y and hence is a Borel set by
the above theorem.
For an x ∈ X we write [x]R for the R-equivalence class of x, i.e.,
[x]R = {y ∈ X : (x, y) ∈ R}.
For a Borel set A ⊆ X we define its saturation
[
[A]R =
[x]R .
x∈A
If R is countable then by Theorem 3.47 the saturation [A]R is Borel since it is the image
of R ∩ (A × X) under the second coordinate projection.
Given a T ∈ JRK, for a Borel function f on R we define T f by T f (x, y) = f (T −1 x, y)
if x ∈ im T and T f (x, y) = 0 otherwise. For a Borel function f on X we define T f by
3.7. THE CONNES-FELDMAN-WEISS THEOREM
121
T f (x) = f (T −1 x) if x ∈ im T and T f (x) = 0 otherwise. For a Borel set A ⊆ X we
write idA for the partial transformation in JRK which has domain A and is the identity
map on A.
D EFINITION 3.48. By a p.m.p. equivalence relation on (X, µ) we mean a countable
Borel equivalence relation R on X such that µ(dom T ) = µ(im T ) for all T ∈ JRK.
D EFINITION 3.49. Let R be a p.m.p. equivalence relation R on (X, µ). A Borel set
A ⊆ X is R-invariant if the symmetric difference of A with its saturation [A]R is a null
set. The relation R is ergodic if every R-invariant Borel subset of X is either null or
conull.
E XAMPLE 3.50. Let G y (X, µ) be a p.m.p. action. We define an equivalence relation R on X by declaring that (x, y) ∈ R if there exists an s ∈ G such that y = sx. This
is called the orbit equivalence relation associated to the action. It is Borel, since it can be
expressed as the union over s ∈ G of the graphs of the Borel automorphisms x 7→ sx and
these graphs are Borel subsets of X × X by the observation following Definition 3.46.
To see that it is a p.m.p. equivalence relation, given any T ∈ JRK take an enumeration
s1 , s2 , . . . of the elements of G and recursively define the sets
An = {x ∈ dom T : T x = sn x}\
n−1
[
Ai ,
i=1
which partition dom T . For each n the set An is Borel, since the set {x ∈ dom T : T x =
sn x} is the inverse image of the diagonal in X ×X under the Borel map dom T → X ×X
given by x 7→ (T x, sn x). Moreover, since the transformation x 7→ sn x is measurepreserving we have µ(An ) = µ(T An ). It follows by countable additivity that µ(dom T ) =
µ(im T ).
Two p.m.p. actions G y (X, µ) and H y (Y, ν) (of possibly different groups) are
said to be orbit equivalent if there are conull sets X ′ ⊆ X and Y ′ ⊆ Y with GX ′ ⊆ X ′
and GY ′ ⊆ Y ′ and a measure isomorphism ϕ : X ′ → Y ′ such that ϕ(Gx) = Hϕ(x) for
every x ∈ X ′ . In other words, the orbit equivalence relations are isomorphic as p.m.p.
equivalence relations, with isomorphism being understood in the analogous way. Given a
p.m.p. action G y (X, µ), the G-invariance of a Borel subset of X clearly only depends
on the orbit equivalence relation, and so ergodicity for p.m.p. actions is invariant under
orbit equivalence. Moreover, a p.m.p. action is ergodic if and only if its orbit equivalence
relation is ergodic.
R EMARK 3.51. By an argument which refines the basic strategy in the proof of
Lemma 3.65 below, Feldman and Moore showed that every countable Borel equivalence
relation is the orbit equivalence relation of a Borel action of a countable group (Theorem 1
of [41]). This action cannot always be taken to be free, however [1].
122
3. AMENABILITY
D EFINITION 3.52. Let R be a p.m.p. equivalence relation on (X, µ). By a partial
subequivalence relation of R we mean a set Q ⊆ R such that Q is an equivalence relation on the set {x ∈ X : (x, x) ∈ Q}, which we denote by Q(0) . If Q and Q′ are
partial subequivalence relations of R, then we say that Q′ is an extension of Q if it can
be written as a disjoint union Q ⊔ Q′′ where Q′′ is a partial subequivalence relation of R.
Partial subequivalence relations will play an important role in our arguments and should
be distinguished from genuine subequivalence relations Q ⊆ R, which can be described
as partial subequivalence relations Q ⊆ R for which Q(0) = X.
Let R be a p.m.p. equivalence relation on (X, µ). We define on R the σ-finite measure
Z
|Ax | dµ(x).
ν(A) =
X
To see that ν is indeed well defined and a σ-finite measure, first observe that, by Theorem 3.47 the relation R is a countable union of Borel graphs, and by applying Theorem 3.47 again to each of these graphs with the roles of the coordinates in X ×X switched
we can produce a partition {Bn } of R into countably many graphs of partial transformations. For each n the restriction fn of the first coordinate projection map X × X → X
to Bn is injective and hence a Borel isomorphism onto its image by Theorem 3.47 (or
more directly by Corollary 15.2 of [82]), and so νn (A) = µ(fn (A ∩ Bn )) defines a finite
Borel measure νn on R. For each Borel set A ⊆ R the function x 7→ |Ax | on X is equal
P
to n 1fn (Bn ) and hence is Borel. Thus ν is well defined and, being a countable sum of
finite measures with disjoint supports, is a σ-finite measure. Since the sets Bn are graphs
of partial transformations we see that ν can also be expressed by taking cross sections in
the other coordinate direction, i.e.,
Z
|Ay | dµ(y).
ν(A) =
X
3.7.2. Amenability, hyperfiniteness, and Reiter’s property. The following properties for R are what the Connes-Feldman-Weiss theorem (Theorem 3.68) asserts to be
equivalent. We say that an equivalence relation is finite if each of its equivalence classes
is finite.
D EFINITION 3.53. (i) The relation R is amenable if there is a unital positive linear
map ϕ : L∞ (R, ν) → L∞ (X) such that ϕ(T f ) = T ϕ(f ) for all T ∈ JRK and f ∈
L∞ (R, ν).
(ii) The relation R is hyperfinite if it is equal a.e. to the union of an increasing sequence
of finite Borel subequivalence relations.
(iii) The relation R satisfies Reiter’s property if there is a sequence of nonnegative
Borel functions λn : R → R such that for each y ∈ X the function λnx on [y]R given by
3.7. THE CONNES-FELDMAN-WEISS THEOREM
123
x 7→ λn (x, y) is a norm-one element of ℓ1 ([y]R ) and kλny − λny′ k1 → 0 as n → ∞ for a.e.
(y, y ′ ) ∈ R.
Note that in the definition of hyperfiniteness we could replace “finite Borel subequivalence relation” by “finite Borel partial subequivalence relation”, since we can always add
the diagonal of X to a partial subequivalence relation to create a genuine subequivalence
relation. This is analogous to unitizing a ∗ -subalgebra of a von Neumann algebra.
E XAMPLE 3.54. The orbit equivalence relation of a p.m.p. action G y (X, µ) of
an amenable group is readily seen to satisfy Reiter’s property using the corresponding
property for groups (see Definition 3.3(ii) and Theorem 3.4). Indeed take a sequence
{fn } of nonegative norm-one functions in ℓ1 (G) witnessing Reiter’s property for G and
P
set λn (x, y) = s∈G, x=sy fn (s−1 ).
Before entering into the main part of the proof of the Connes-Feldman-Weiss theorem
in the next section, we collect a couple of facts about amenability and hyperfiniteness.
The first is that hyperfiniteness implies that the relation is the orbit equivalence relation
of a p.m.p. Z-action, which is the implication (iii)⇒(v) in Theorem 3.68. We will in
fact prove a purely Borel-theoretic statement which implies its measured version. The
converse is also true, but we will not need it.
L EMMA 3.55. Let R be a countable Borel equivalence relation on X such that R =
S∞
n=1 Rn for some increasing sequence R1 ⊆ R2 ⊆ . . . of finite Borel subequivalence
relations. Then R is the orbit equivalence relation of a Borel Z-action.
P ROOF. Throughout the proof π1 and π2 denote the first and second coordinate projections X ×X → X. We will repeatedly and tacitly apply Theorem 3.47 in this situation.
Fix a Borel linear order <′ on X, which we can do by identifying X as a Borel space
with [0, 1] or a subset of N (Theorem A.17). For a given n, let fn : X → X be the
map which assigns to a point the <′ -least element in its Rn class. Then fn is Borel,
for if A is a Borel subset of X then, writing <′ in set notation as Q, the set fn−1 (A) =
π2 (Rn ∩ ((A\π2 (Rn ∩ Q)) × X))) is Borel.
We may assume that R1 is the diagonal in X ×X. Then, given distinct x, y in the same
R class, we can take the largest n such that x and y lie in different Rn classes and declare
that x < y if fn (x) <′ fn (y). This defines a linear order < on each R class, and this order
is discrete (i.e., only finitely many elements lie between any two given elements), for if x,
y, and n are as in the definition then the inequalities x < z < y can only occur if z lies in
the same Rn+1 class as x and y, which is finite. Thus every R class is order isomorphic
to one of Z, N, −N, or {1, . . . , n} for some n ∈ N. We partition X accordingly into Rinvariant sets B, B+ , B− , and Bn for n ∈ N. We will next argue that these are all Borel,
124
3. AMENABILITY
which enables us to reduce the verification of the lemma to the restriction of R to each of
these subsets individually, since Borel subsets of X are standard (Corollary A.14).
Observe that the set O := {(x, y) ∈ R : x < y} is Borel, for it can be written as
S∞
−1
′
n=1 [(Rn+1 \Rn ) ∩ (fn × fn ) (Q)] where Q stands for < as before. For each n ≥ 1 the
set Cn of all points in X which are nth least in their R class can be recursively expressed
as the complement in X of the set
n−1
n−1
[ [
X\
∪
π2 O ∩
Cj × X
Cj
j=1
j=1
and hence is Borel. Then for each n ≥ 1 the set
Bn = π1 (R ∩ (X × Cn ))\π1 (R ∩ (X × Cn+1 ))
is Borel, as is
B+ =
∞
\
n=1
π1 (R ∩ (X × Cn )).
The set B− is similarly seen to be Borel by considering nth greatest elements, and so B
is Borel as well.
Now to derive the conclusion of the lemma in the case of B+ , let g : N → N be
the bijection which is conjugate to the predecessor map on Z under some set-theoretic
identification of N and Z. Then we define a bijection ϕ : B+ → B+ by sending the nth
point in an R class to the g(n)th point in the same class, as ordered by <. Then the orbits
of ϕ are precisely the R classes over B+ . Also, ϕ is Borel, for if A ⊆ X then
−1
ϕ (A) =
∞
[
n=1
−1
ϕ (B+ ∩ Cn ∩ A) =
∞
[
n=1
B+ ∩ Cg−1 (n) ∩ π1 (R ∩ (X × (Cn ∩ A))).
Similarly, ϕ−1 is Borel, and so ϕ defines a Borel Z-action. We can argue similarly for
B− , as well as for Bn , in which case we take g to be a cyclic permutation of {1, . . . , n}.
It remains to treat B. The difference here is that we do not have Borel transversals
like the Cn . Define the bijection ϕ : B → B by sending a point to its predecessor under <
in the same R class. It remains to verify that ϕ is a Borel automorphism. Write D for the
image of the Borel set (O × X) ∩ (X × O) under the projection X × X × X× → X × X
onto the first and third coordinates. Then D is equal to the set of all (x, z) ∈ X × X such
that there exists a y ∈ X satisfying x < y < z, and it is Borel by Theorem 3.47 since the
sections of (O × X) ∩ (X × O) through the second coordinate are countable. Now given
a Borel set A ⊆ B we have
((A × X) ∩ O)\D = {(x, z) ∈ A × X : x < z and there is no y ∈ X with x < y < z}
3.7. THE CONNES-FELDMAN-WEISS THEOREM
125
and the projection of this set onto the second coordinate, which is a Borel set by Theorem 3.47, is equal to ϕ−1 (A). Thus ϕ is a Borel map, and a similar argument shows that
ϕ−1 is also Borel, as desired.
Secondly, we establish in Lemma 3.62 a partial inheritance property for a condition
which is weaker than amenability and involves invariant states on L∞ (R, ν) instead of
the equivariant maps L∞ (R, ν) → L∞ (X) in Definition 3.53(i). It will follow from the
Connes-Feldman-Weiss theorem that amenability itself passes to Borel partial subequivalence relations supported on nonnull sets (with the measure renormalized) since hyperfiniteness clearly does, but in order to prove the Connes-Feldman-Weiss theorem we will
need to use Lemma 3.62 to leverage amenability in a local way so as to be able to apply a
maximality argument.
Let R be a p.m.p. equivalence relation on (X, µ). By definition, R is amenable if there
is a unital positive linear map ϕ : L∞ (R, ν) → L∞ (X) such that ϕ(T f ) = T ϕ(f ) for all
T ∈ JRK and f ∈ L∞ (R, ν). InR this case we can define a unital positive linear functional
σ : L∞ (R, ν) → C by σ(f ) = X ϕ(f ) dµ. This linear functional is invariant in the sense
that
σ(T f ) = σ(iddom T f )
for all T ∈ JRK and f ∈ L∞ (R, ν).
D EFINITION 3.56. A unital positive linear functional L∞ (R, ν) → C possessing the
above invariance property will be called an invariant state on L∞ (R, ν).
Asking for the existence of an invariant state is thus a weakening of amenability, and
this is the subject of Lemma 3.62, which we will derive from the following series of
lemmas.
For Borel sets A, B ⊆ X we define
RA = R ∩ (A × A)
and
RA,B = R ∩ (A × B).
When A is nonnull we can view RA as a p.m.p. equivalence relation by normalizing
the measure of A to be 1, and so in particular we can speak about invariant states on
L∞ (RA , µ(A)−1 ν). We say that two Borel sets A, B ⊆ X are equivalent if there is a
T ∈ JRK such that dom T = A and im T = B.
L EMMA 3.57. Let A and B be Borel subsets of X. If µ([B]R ∩ A) > 0, then there is
a nonnull Borel subset of B which is equivalent to a subset of A.
P ROOF. Applying Theorem 3.47 twice as in the definition of ν at the end of the previous section, we can find a countable partition {Rn } of RA,B such that each Rn is the
S
graph of some Tn ∈ JRK. Then [B]R ∩ A = ∞
n=1 dom Tn . If µ([B]R ∩ A) > 0, then
126
3. AMENABILITY
µ(dom Tn ) > 0 for some n ∈ N, and imTn ⊆ B has measure µ(dom Tn ) and is equivalent
to dom Tn ⊆ A.
L EMMA 3.58. Let A and B be Borel subsets of X. Then there exists a Borel set
B1 ⊆ B such that B1 is equivalent to a subset of A and µ([B\B1 ]R ∩ A) ≤ µ(B).
P ROOF. Consider the family of collections of Borel subsets {Rj }j∈J of RA,B such
that for each j ∈ J the relation Rj is nonnull and equal to the graph of some Tj ∈ JRK,
and each of the collections {dom Tj }j∈J and {im Tj }j∈J is pairwise disjoint. We define a
partial order on this family by declaring that {Rj }j∈J {Rj′ ′ }j ′ ∈J ′ if J ⊆ J ′ and Rj = Rj′
for all j ∈ J. By Zorn’s Lemma this family has a maximal element {Rj }j∈J . Set A1 =
S
S
S
j∈J Rj is the graph of some element of
j∈J im Tj . Note that
j∈J dom Tj and B1 =
JRK and hence A1 and B1 are equivalent. By Lemma 3.57, if µ([B\B1 ]R ∩ (A\A1 )) > 0
then there is a nonnull Borel subset of B\B1 which is equivalent to a subset of A\A1 , say
via T ∈ JRK. Adding the graph of T to {Rj }j∈J contradicts the maximality of {Rj }j∈J .
Thus µ([B\B1 ]R ∩ (A\A1 )) = 0. Therefore µ([B\B1 ]R ∩ A) ≤ µ(A1 ) = µ(B1 ) ≤
µ(B).
L EMMA 3.59. Let A and B be Borel subsets of X with B ⊆ [A]R . Let ε > 0. Then
there is a finite collection {Bj }j∈J of pairwise disjoint Borel subsets of B such that each
S
Bj is equivalent to a subset of A and µ(B\ j∈J Bj ) < ε.
P ROOF. Consider the family of collections of pairwise disjoint Borel subsets {Bj }j∈J
of B such that for each j ∈ J the set Bj is nonnull and equivalent to a subset of A. We
define a partial order on this family by declaring that {Bj }j∈J {Cj ′ }j ′ ∈J ′ if J ⊆ J ′ and
Bj = Cj for all j ∈ J. By Zorn’s Lemma this family has a maximal element {Bj }j∈J .
S
S
By Lemma 3.57, if µ(B\ j∈J Bj ) > 0 then B\ j∈J Bj has a nonnull Borel subset
which is equivalent to a subset of A, contradicting the maximality of {Bj }j∈J . Thus
S
µ(B\ j∈J Bj ) = 0. Since each Bj is nonnull and the sets Bj are pairwise disjoint, J is
S
countable. Take a finite subset J ′ of J such that µ( j∈J\J ′ Bj ) < ε. Then
[
[ Bj < ε.
Bj = µ
µ B\
j∈J ′
j∈J\J ′
L EMMA 3.60. Let A and B be Borel subsets of X. Then there is a sequence {Bn } of
pairwise disjoint Borel subsets of B such that each Bn is equivalent to a subset of A and
S
limn→∞ µ([B\ nm=1 Bm ]R ∩ A) = 0.
P ROOF. Replacing B with [A]R ∩ B if necessary, we may assume that B ⊆ [A]R .
We show first that for any ε > 0 there is a finite collection {Bi }i∈I of pairwise disjoint
S
Borel subsets of B such that each Bi is equivalent to a subset of A and µ([B\ i∈I Bi ]R ∩
A) < ε. By Lemma 3.59 we can find a finite collection {Bj }j∈J of pairwise disjoint Borel
3.7. THE CONNES-FELDMAN-WEISS THEOREM
127
S
subsets of B such that each Bj is equivalent to a subset of A and µ(B\ j∈J Bj ) < ε. By
S
Lemma 3.58 we can find a Borel subset C of B\ j∈J Bj such that C is equivalent to a
subset of A and
[ [ ∩ A ≤ µ B\
Bj < ε.
µ B\ C ∪
Bj
j∈J
R
j∈J
Now we may set {Bi }i∈I = {Bj }j∈J ∪ {C}.
By a recursive procedure, for each k = 1, 2, . . . we find a finite collection {Bi }i∈Ik of
S
S
pairwise disjoint Borel subsets of B\ k−1
j=1
i∈Ij Bi such that each Bi is equivalent to a
Sk S
P
subset of A and µ([B\ j=1 i∈Ik Bi ]R ∩ A) < 1/k. Now we set nk = kj=1 |Ij | and list
{Bi }i∈Ik as Bnk−1 +1 , . . . , Bnk .
L EMMA 3.61. Let σ : L∞ (R, ν) → C be an invariant state satisfying |σ(idB f )| ≤
µ(B)kf k for all Borel sets B ⊆ X and f ∈ L∞ (R, ν). Let A be a Borel subset of X with
σ(1RA ) = 0. Then σ(1RX,A ) = 0.
P ROOF. Set B = [A]R . Let ε > 0. By Lemma 3.59 we can find a finite collection
{Bj }j∈J of pairwise disjoint Borel subsets of B such that each Bj is equivalent to a subset
S
S
of A and µ(B\ j∈J Bj ) < ε. Set D = B\ j∈J Bj . For each j ∈ J take Tj ∈ JRK with
im Tj = Bj and dom Tj ⊆ A. Then
0 ≤ σ(1RBj ,A ) = σ(iddom Tj−1 1RBj ,A ) = σ(Tj−1 1RBj ,A ) ≤ σ(1RA ) = 0,
where the two inequalities come from the linearity and positivity of σ. Thus σ(1RBj ,A ) =
0. Then
X
σ(1RX,A ) = σ(1R[A]R ,A ) = σ(1RD,A ) +
σ(1RBj ,A ) = σ(1RD,A ) ≤ µ(D) < ε,
j∈J
where the nonstrict inequality is from our hypothesis on σ. Since ε is arbitrary, we conclude that σ(1RX,A ) = 0.
L EMMA 3.62. Suppose that there is an invariant state σ : L∞ (R, ν) → C satisfying
|σ(idB f )| ≤ µ(B)kf k for all Borel sets B ⊆ X and f ∈ L∞ (R, ν). Let A be a nonnull
Borel subset of X. Then RA has an invariant state.
P ROOF. Set B = X\A. By Lemma 3.60 there is a sequence {Bn } of pairwise disjoint nonnull Borel subsets of B such that each Bn is equivalent to a subset of A and
S
limn→∞ µ([B\ nm=1 Bm ]R ∩ A) = 0.
If σ(1RA ) > 0, then we can restrict σ to L∞ (RA , µ(A)−1 ν) and multiply it by 1/σ(1RA )
to get an invariant state for RA . We may thus assume that σ(1RA ) = 0.
For each n ∈ N take an An ⊆ A to which Bn is equivalent. It is easy to see that
A = An ⊔ (A\An ) is equivalent to Bn ⊔ (A\An ), so that RA is isomorphic to RBn ∪(A\An ) .
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3. AMENABILITY
If σ(1RBn ) > 0 for some n, then σ(1RBn ∪(A\An ) ) ≥ σ(1RBn ) > 0, so that as in the previous
paragraph we can produce an invariant state for RBn ∪(A\An ) and hence for RA . Therefore
we may assume that σ(1RBn ) = 0 for all n.
S
Next we claim that σ(1RA,X ) = 0. Let ε > 0. Take an n such that µ([B\ nm=1 Bm ]R ∩
A) < ε. By Lemma 3.61 we have σ(1RX,Bm ) = 0 for all m. Since σ is positive, we get
σ(1RA,Bm ) = 0 for all m. Then
σ(1RA,X ) = σ 1RA,B\ Sn
m=1 Bm
= σ 1RA,B\ Sn
m=1 Bm
+ σ(1RA ) +
n
X
σ(1RA,Bm )
m=1
= σ 1R[B\ Sn Bm ]R ∩A,B\ Sn Bm
m=1
m=1
n
[
≤ µ B\
Bm ∩ A < ε,
m=1
R
where the nonstrict inequality is from our hypothesis on σ. Since ε is arbitrary, we conclude that σ(1RA,X ) = 0.
Finally we obtain the contradiction
1 = σ(1R ) = σ(1RA,X ) + σ(1RB,X ) = σ(1RB,X ) ≤ µ(B) < 1,
where the nonstrict inequality follows from our hypothesis on σ.
R EMARK 3.63. In general, the existence of an invariant state for R is not equivalent
to amenability. For example, consider the case R = RA ∪ RX\A for some Borel set
A ⊆ X with 0 < µ(A) < 1. If RA is amenable, then the composition L∞ (R, ν) →
L∞ (RA , µ(A)−1 µ) → C yields an invariant state for R, although R need not be amenable.
However, the two properties are equivalent when R is ergodic, which can be seen as
follows using the Connes-Feldman-Weiss theorem. If µ is not atomless and R is ergodic,
then modulo null sets X consists of finitely many points with uniform distribution and
R is equal to X × X, and so R is amenable. So assume that µ is atomless and that R
is ergodic and possesses an invariant state. To show that R is amenable, by the proof
of (i)⇒(ii) of Theorem 3.68 we just need to show that for every nonnull Borel subset
A ⊆ X the relation RA has an invariant state. Take n ∈ N with µ(A) ≥ 1/n. Since µ is
atomless, by Theorem A.20 we can find a Borel partition A1 , . . . , An of X with A1 ⊆ A
P
and µ(Ai ) = 1/n for all i. Since 1 = σ(1R ) = ni=1 σ(1RX,Ai ), one has σ(1RX,Aj ) > 0
for some j. Since R is ergodic, using a maximality argument one can show that each Ai
is equivalent to Aj . Thus
σ(1RX,Aj ) =
n
X
i=1
σ(1RAi ,Aj ) = nσ(1RAj )
3.7. THE CONNES-FELDMAN-WEISS THEOREM
129
and hence σ(1RAj ) > 0. Using the equivalence between Aj and A1 , one sees that Aj \A
is equivalent to some B ⊆ A1 . Then A = B ⊔ (A\B) is equivalent to Aj ∪ (A\B) =
(Aj \A) ⊔ (A\B), and hence RA is isomorphic to RAj ∪(A\B) . Note that σ(1RAj ∪(A\B) ) ≥
σ(1RAj ) > 0. Normalizing σ then produces an invariant state for RAj ∪(A\B) , and hence
for RA .
3.7.3. The Connes-Feldman-Weiss theorem. We now embark on a series of lemmas which will be used to show that amenability implies hyperfiniteness, which is the
core implication (i)⇒(iii) of the Connes-Feldman-Weiss theorem, Theorem 3.68. The
aim of these lemmas is to show that if R is amenable then every Borel set K ⊆ R which
is bounded in the sense of Definition 3.64 contains a Borel partial subequivalence relation
Q over a set Q(0) of nonnegative µ-measure such that most pairs in K\Q have coordinates
both lying outside of Q(0) . By applying this fact in a (transfinite) recursive way, we can
then fill up most of K with disjoint Borel partial subequivalence relations whose union is
a subequivalence relation on all of X. This will be the proof of (i)⇒(ii) in Theorem 3.68,
phrased in the Zorn’s lemma language of maximality.
D EFINITION 3.64. We say that a set K ⊆ R is bounded if supx∈X |Kx | and supy∈X |K y |
are both finite.
L EMMA 3.65. Let K be a bounded Borel subset of R. Then K is a finite union of
graphs of partial transformations in JRK.
P ROOF. Set n = supx∈X |Kx | and m = supy∈X |K y |. Recursively applying TheoS
rem 3.47, for k = 1, . . . , n we find a Borel graph Qk ⊆ K\ k−1
i=1 Qi . Then for each
fixed k = 1, . . . , n we recursively apply Theorem 3.47 for j = 1, . . . , m to find Borel sets
Sj−1
Qk,j ⊆ Qk \ i=1
Q which are graphs when we view X × X with its factors swapped.
Sn Smk,i
Then K = k=1 j=1 Qk,j , and the map Tk,j defined by a given Qk,j is a partial transformation. Finally we observe that each Tk,j is Borel, for if A is a Borel subset of X then
−1
(A) is the projection of Qk,j ∩ (X × A) onto the first coordinate in X × X and thus
Tk,j
is a Borel set by Theorem 3.47.
The following is akin to the local characterization of freeness for a group action in
terms of the action on nonnull sets (Proposition 1.4), and the proof is essentially a reduction to that situation using Lemma 3.65.
L EMMA 3.66. Let K be a bounded Borel subset of R and let B be a nonnull subset
of X. Then there exists a nonnull set B ′ ⊆ B such that the sets {x ∈ [y]R : (x, y) ∈ K}
and {x ∈ [y ′ ]R : (x, y ′ ) ∈ K} are disjoint for all distinct y, y ′ ∈ B ′ .
130
3. AMENABILITY
P ROOF. Observe first that, by Proposition A.22, if T is any partial transformation of
X and C is any nonnull subset of X, then we can find a nonnull set C ′ ⊆ C such that one
of the following holds:
(i) C ′ ∩ dom T = ∅,
(ii) C ′ ⊆ dom T and T C ′ ∩ C ′ = ∅,
(iii) C ′ ⊆ dom T and T x = x for all x ∈ C ′ .
S
Now by Lemma 3.65 we can write K = ni=1 gr Ti for some T1 , . . . , Tn ∈ JRK. By
applying the above observation recursively to the partial transformations Tj ◦ Ti−1 for
i, j = 1, . . . , n under some fixed enumeration, we can find a single nonnull set B ′ ⊆ B
such that for each of the Tj ◦ Ti−1 one of the above three conditions holds. Then for all
distinct y, y ′ ∈ B ′ we have Ti−1 y 6= Tj−1 y ′ for all i, j = 1, . . . , n, and so B ′ satisfies the
requirements.
L EMMA 3.67. Suppose that R admits an invariant state (Defintion 3.56). Let K be a
bounded Borel subset of R and ε > 0. Then there is a finite Borel partial subequivalence
relation Q ⊆ R with µ(Q(0) ) > 0 such that
ν({(x, y) ∈ K\Q : x ∈ Q(0) or y ∈ Q(0) }) < εµ(Q(0) ).
P ROOF. Viewing L1 (R, ν) as a subspace ofRL∞ (R, ν)∗ under the identification of a
function g ∈ L1 (R, ν) with the functional f 7→ R gf dν, we claim that σ is in the weak∗
1
closure of the convex set L1 (R, ν)+
1 of all nonnegative functions in L (R, ν) of norm one.
If this is not the case, then by the Hahn-Banach theorem there are an f ∈ L∞ (R, ν) and
an α > 0 such that
Z
gf dν
re
R
+ α ≤ re σ(f )
for all g ∈ L1 (R, ν)+
1 . Taking the real part of f and adding a constant function if
necessary, we may assume that f is real-valued and f ≥ 0. Take a set A ⊆ R with
0 < ν(A) < ∞ such that f (x, y) ≥ kf k − α/2 for all (x, y) ∈ A. Then
Z
Z
α
−1
ν(A)−1 1A f dν + ,
ν(A) 1A f dν + α ≤ σ(f ) ≤ kf k ≤
2
R
R
a contradiction which proves the claim.
By Lemma 3.65 there exist partial transformations T1 , . . . , Tn ∈ JRK such that K =
Sn
1
1
⊕n
of n copies of L1 (R, ν) as a subi=1 gr Ti . Regard the ℓ -direct sum L (R, ν)
space of the dual of the ℓ∞ -direct sum L∞ (R, ν)⊕n via the canonical embedding. Then
the weak∗ topology on this dual restricts to the weak topology on L1 (R, ν)⊕n . Since
σ(Ti f − iddom Ti f ) = 0 for every i = 1, . . . , n and f ∈ L∞ (R, ν), the claim from the
previous paragraph implies that the subset {(Ti g − iddom Ti−1 g)ni=1 : g ∈ L1 (R, ν)+
1 } of
1
⊕n
L (R, ν) contains 0 in its weak closure. Since this subset is convex, its weak and norm
3.7. THE CONNES-FELDMAN-WEISS THEOREM
131
closures coincide by Mazur’s theorem, and so given an ε > 0 we can find a particular
g ∈ L1 (R, ν)+
1 such that kTi g − iddom Ti−1 gk1 < ε/n for every i = 1, . . . , n. We may
assume that supp g is bounded by perturbing g slightly if necessary.
∞), for t, t′ ≥ 0 we observe that t =
R ∞ Writing Ea for the ′ indicator
R ∞ function of [a,
Ea (t) da and |t − t | = 0 |Ea (t) − Ea (t′ )| da. Thus, writing ga for the function on
0
R which takes value 1 or 0 at a point (x, y) depending on whether or not g(x, y) ≥ a, we
have, for all x ∈ dom Ti−1 and y ∈ X,
Z ∞
ga (x, y) da = g(x, y),
0
Z ∞
|ga (Ti−1 x, y) − ga (x, y)| da = |g(Ti−1 x, y) − g(x, y)|,
0
so that, using Fubini’s theorem,
Z ∞X
n Z
−1
|ga (Ti x, y) − ga (x, y)| · 1dom Ti−1 (x) dν da
0
i=1
R
=
=
=
n Z Z
X
i=1 R
n Z
X
i=1
n
X
i=1
R
∞
0
|ga (Ti−1 x, y) − ga (x, y)| · 1dom Ti−1 (x) da dν
|g(Ti−1 x, y) − g(x, y)| · 1dom Ti−1 (x) dν
kTi g − iddom Ti−1 gk1
<ε=ε
Z Z
R
∞
ga (x, y) da dν =
0
Z
∞
0
Z
ε ga (x, y) dν da.
R
We can thus find a particular a > 0 for which the bracketed expression at the end of this
display is larger than the one at the beginning. In other words, using the definition of the
measure ν,
Z X
n
X
−1
|ga (Ti x, y) − ga (x, y)| · 1dom Ti−1 (x) dµ(y)
X
i=1 x∈[y]R
<ε
Z X
X
ga (x, y) dµ(y).
x∈[y]R
Hence the set A of all y ∈ X satisfying
(21)
n
X
X
i=1 x∈[y]R
|ga (Ti−1 x, y) − ga (x, y)| · 1dom Ti−1 (x) < ε
X
x∈[y]R
ga (x, y)
132
3. AMENABILITY
has nonzero measure.
Since supp g is bounded, by Lemma 3.66 there is a nonnull set A′ ⊆ A such that the
sets {x ∈ [y]R : (x, y) ∈ supp g} and {x ∈ [y ′ ]R : (x, y ′ ) ∈ supp g} are disjoint for all
distinct y, y ′ ∈ A′ . For every y ∈ A′ define the Borel set
Fy = {x ∈ [y]R : ga (x, y) = 1}.
These sets are finite since for each y ∈ X the function x 7→ ga (x, y) has finite support,
and they are pairwise disjoint by our choice of A′ . Hence the set
[
{(x, w) : x, w ∈ Fy }
Q=
y∈A′
S
is a finite partial subequivalence relation, with Q(0) = y∈A′ Fy .
For every y ∈ A′ and i = 1, . . . , n, write Biy for the set of all x ∈ [y]R ∩dom Ti−1 such
that Fy contains one of Ti−1 x and x but not both, i.e., such that |ga (Ti−1 x, y) − ga (x, y)| =
S
1. Then, setting Bi = y∈A′ Biy and using (21),
n
n Z
X
X
|Biy | dµ(y)
µ(Bi ) ≤
i=1
=
i=1 A
n Z
X
i=1
<ε
=ε
Z
Z
′
X
A′ x∈[y]
R
X
|ga (Ti−1 x, y) − ga (x, y)| · 1dom Ti−1 (x) dµ(y)
ga (x, y) dµ(y)
A′ x∈[y]
R
A′
|Fy | dµ(y) = εµ(Q(0) ).
This shows in particular that µ(Q(0) ) > 0.
Now each element of K is of the form (Ti−1 x, x) for some i = 1, . . . , n and x ∈
dom Ti−1 , and if Ti−1 x ∈ Fy or x ∈ Fy for some y ∈ A′ , then either (Ti−1 x, x) ∈ Q or
x ∈ Biy . Setting
D = {(x, y) ∈ K\Q : x ∈ Q(0) or y ∈ Q(0) }
P
it follows that ν(D) ≤ ni=1 µ(Bi ) < εµ(Q(0) ).
Condition (ii) in the following theorem is the analogue of the Følner property for
equivalence relations. The partial subequivalence relation Q plays the role of a Følner
set. By Lemma 3.65, the set K is a finite union of partial transformations, and these
play the role of the finite set of group elements which serve to quantitify the approximate
invariance of a given Følner set. In this case the approximate invariance of a Følner set
translates as the approximate containment of K in Q.
3.7. THE CONNES-FELDMAN-WEISS THEOREM
133
T HEOREM 3.68. The following are equivalent:
(i) R is amenable,
(ii) for every bounded Borel set K ⊆ R and ε > 0 there is a finite Borel partial
subequivalence relation Q ⊆ R such that ν(K\Q) ≤ ε,
(iii) R is hyperfinite,
(iv) R has Reiter’s property,
(v) R is a.e. the orbit equivalence relation of a measure-preserving transformation
of (X, µ).
P ROOF. (i)⇒(ii). We apply a maximality argument, much like in our proofs of the
Rokhlin lemma in both the integer and general amenable cases (Lemma 3.72 and Theorem 3.43). Let K be a bounded Borel subset of R and let ε > 0. Form the collection C
of pairs (Q, J) where Q is a finite Borel partial subequivalence relation of R and J is a
Borel subset of K for which
(1) ν(K\J) ≤ εµ(Q(0) ),
(2) for a.e. (x, y) ∈ J, if x ∈ Q(0) or y ∈ Q(0) then (x, y) ∈ Q.
We define an order on C by declaring (Q, J) (Q′ , J ′ ) to mean that Q′ is an extension
of Q (Definition 3.52) and J ′ ⊆ J.
Now given any totally ordered subcollection C′ of C, take a sequence {(Qn , Jn )}∞
n=1
S
T∞
(0)
in C′ such that supn∈N µ(Qn ) = sup(Q,J)∈C′ µ(Q(0) ). Then ( ∞
Q
,
J)
is
easily
n=1 n
n=1
seen to be an element of C which is an upper bound for C′ . Since C contains (∅, K) and
hence is nonempty, it follows by Zorn’s lemma that C has a maximal element (Q, J).
To obtain (ii) it is enough to show that µ(Q(0) ) = 1, for then by (1) and (2) we have
ν(K\J) ≤ ε and ν(J\Q) = 0, so that ν(K\Q) ≤ ε. Suppose that µ(Q(0) ) 6= 1. Set
A = X\Q(0) . Since R is amenable, by the discussion before Lemma 3.57 it admits an
invariant state which satisfies the hypothesis of Lemma 3.62, so that RA has an invariant
state. We can then apply Lemma 3.67 using the bounded Borel set JA = J ∩ (A × A) to
(0)
obtain a finite Borel partial subequivalence relation Q1 ⊆ R ∩ (A × A) with µ(Q1 ) > 0
such that the set
(0)
(0)
L = {(x, y) ∈ JA \Q1 : x ∈ Q1 or y ∈ Q1 }
(0)
satisfies ν(L) < εµ(Q1 ). Define Q′ = Q ∪ Q1 and J ′ = J\L. Then Q′ is a finite Borel
partial subequivalence relation and
(0)
ν(K\J ′ ) = ν(K\J) + ν(J\J ′ ) ≤ εµ(Q(0) ) + εµ(Q1 ) = εµ(Q′(0) ).
Now suppose that (x, y) ∈ J ′ and one of x and y lies in Q′(0) . If x ∈ Q′(0) then we have
the following three possibilities: (i) x ∈ Q(0) , in which case (x, y) ∈ Q since (Q, J) ∈ C
(0)
(0)
and J ′ ⊆ J, (ii) x ∈ Q1 and y ∈ A, in which case (x, y) ∈ Q1 ⊆ Q′ , (iii) x ∈ Q1 and
134
3. AMENABILITY
y∈
/ A, in which case y ∈ Q(0) and hence (x, y) ∈ Q as in (i). By similar observations, if
y ∈ Q′(0) then (x, y) belongs to Q′ . We have thus verified that (Q′ , J ′ ) is a member of C.
Now since Q′ is an extension of Q by construction, we have (Q, J) (Q′ , J ′ ), which
contradicts maximality. It must therefore be the case that ν(Q(0) ) = 1, as desired.
(ii)⇒(iii). By Theorem 3.47, there is a countable collection {Ti : i ∈ N} of partial
S
Sn
transformations in JRK such that R = ∞
we obtain an
i=1 gr Ti . Setting Kn =
i=1 gr T
Si ∞
increasing sequence K1 ⊆ K2 ⊆ . . . of bounded subsets of R such that n=1 Kn = R.
By (ii) we can find finite Borel partial subequivalence relations Q1 , Q2 , . . . of R such that
T
ν(Kn \Qn ) < 2−n for every n. Declaring Rn to be the union of ∞
k=n Qk and the diagonal
of X × X we obtain a finite subequivalence relation of R such that
ν(Kn \Rn ) ≤
∞
X
j=n
ν(Kj \Qj ) <
∞
X
2−j = 2−n+1 .
j=n
We deduce that R is equal a.e. to the union of the increasing sequence R1 ⊆ R2 ⊆ . . . ,
yielding (iii).
(iii)⇒(v). By neglecting a conull set, we may assume that R is in fact equal to
S∞
n=1 Rn for some finite Borel subequivalence relations R1 ⊆ R2 ⊆ . . . . It follows by
Lemma 3.55 that R is the orbit equivalence relation of a Borel Z-action, and this action is
p.m.p. by the definition of a p.m.p. equivalence relation.
(v)⇒(iv). This is a special case of Example 3.54 by viewing a transformation as a
Z-action.
(iv)⇒(i). Let {λn } be a sequence witnessing Reiter’s property. For each n define the
unital positive linear map ϕn : L∞ (R, ν) → L∞ (X, µ) by
ϕn (f )(x) =
X
λnx (y)f (x, y).
y∈[x]R
The space of unital positive linear maps L∞ (R, ν) → L∞ (X, µ) ∼
= L1 (X, µ)∗ is compact
under the topology of pointwise weak∗ convergence, and so the sequence {ϕn } has a
cluster point ϕ. By the Reiter condition, given T ∈ JRK and ε > 0 we can find a Borel
set X0 ⊆ X with µ(X0 ) as close to 1 as we wish such that for all large enough n we
have kλnx − λnT −1 x k1 < ε for all x ∈ X0 ∩ dom T −1 . Then for all f ∈ L∞ (R, ν) and a.e.
x ∈ X0 ∩ dom T −1 we have
|(ϕn (T f ) − T ϕn (f ))(x)| ≤
X
y∈[x]R
|λnx (y) − λnT −1 x (y)| · |f (T −1 x, y)|
≤ kλnx − λnT −1 x k1 kf k∞ ≤ εkf k∞ .
3.7. THE CONNES-FELDMAN-WEISS THEOREM
135
Hence for any g ∈ L1 (X, µ) we have, by demanding that the measure of X0 to be close
enough to 1 to ensure that kg · 1X\X0 k1 < ε,
Z
(ϕn (T f ) − T ϕn (f ))g dµ ≤ 2kf k∞ kg · 1X\X0 k1 + εkf k∞ kg · 1X0 k1
X
≤ εkf k∞ (2 + kgk1 ).
As we can choose n so that ϕ(f ) and ϕ(T f ) areRapproximated in the weak∗ topology by
ϕn (f ) and ϕn (T f ), respectively, this shows that X (ϕ(T f ) − T ϕ(f ))g dµ is in fact equal
to zero, from which we conclude that ϕ(T f ) = T ϕ(f ). Hence R is amenable.
As an application of Theorem 3.68, we give an ergodic-theoretic characterization of
amenability in Theorem 3.71 below.
D EFINITION 3.69. Let G y (X, µ) be a p.m.p. action. A sequence {An } of Borel
subsets of X is said to be asymptotically invariant if limn→∞ µ(An ∆sAn ) = 0 for all
s ∈ G, and such a sequence is said to be trivial if limn→∞ µ(An )(1 − µ(An )) = 0.
The action is strongly ergodic if every asymptotically invariant sequence of Borel sets is
trivial.
The asymptotic invariance of the sequence {An } depends only on the orbit equivalence class of the action. Indeed suppose that α : G y (X, µ) and β : H y (X, µ)
are orbit equivalent actions and {An } is asymptotically invariant for β. Let s ∈ G. Then
there is a Borel partition {Dt }t∈H of X such that αs agrees with βt on Dt , for we can fix
an enumeration t1 , t2 , . . . of H and for n ∈ N set Cn = {x ∈ X : αs x = βtn x}, which
S
is a Borel set by Proposition A.21, and then recursively define Dtn = Cn \ n−1
m=1 Cm
(which is just Cn if β is free). Now given an ε > 0 take a finite set F ⊆ H such that
S
µ( t∈H\F Dt ) < ε/4. Using the fact that limn→∞ µ(βt (Dt ∩ An )\An ) = 0 for all t ∈ F
by asymptotic invariance, we thus see for all large enough n we have µ(αs An \An ) < ε/2
and hence µ(αs An ∆An ) < ε. Thus {An } is asymptotically invariant for α.
Since the invariance of a measurable set under a p.m.p. action clearly only depends
on the orbit equivalence class of the action, we immediately obtain the following.
P ROPOSITION 3.70. Strong ergodicity is invariant under orbit equivalence.
T HEOREM 3.71. The group G is amenable if and only if no ergodic p.m.p. action on
an atomless probability space is strongly ergodic. One can also replace “ergodic” by
“weak mixing”, and even “Bernoulli”.
P ROOF. Suppose first that G is amenable. Let G y (X, µ) be an ergodic p.m.p.
action on an atomless probability space. By Example 3.54 and Theorem 3.68, this action
is orbit equivalent to a measure-preserving transformation T : X → X. Since ergodicity
136
3. AMENABILITY
is an invariant of orbit equivalence by Proposition 3.70, T is ergodic. It follows that T is
free, since for every n ∈ Z the set {x ∈ X : T n x = x} is Borel by Proposition A.21
and hence has measure zero by ergodicity and Theorem A.20. Using the Rokhlin lemma
(Lemma 3.72), for every n ∈ N we find a Borel set B ⊆ X such that the sets T k B
S2n−1 k
for k = 0, . . . , 2n − 1 are pairwise disjoint and µ( k=0
T B) ≥ 1 − 1/n, and set
Sn−1 k
An = k=0 T B. Then {An } is a nontrivial asymptotically invariant sequence for T and
hence also for the original action by the observation before Proposition 3.70.
In the converse direction, suppose that G y (X, µ) is a nontrivial Bernoulli action
admitting a nontrivial asymptotically invariant sequence {An } of Borel sets, which we
may assume to satisfy inf n∈N µ(An )(1 − µ(An )) > 0. Then we can form the unit vectors
fn := (1An − µ(An )1X )/k1An − µ(An )1X k2 in L2 (X) ⊖ C1, and for the Koopman
representation κ we have limn→∞ kκ(s)fn − fn k2 = 0 for all s ∈ G given that the
quantities k1An − µ(An )1X k2 have a nonzero uniform lower bound by our assumption on
the sequence {An }. As described in the discussion on Bernoulli actions at the beginning
of Section 1.3, the restriction of the Koopman representation to L2 (X) ⊖ C1 is contained
in λ⊕I
G for some countable index set I, and so the latter representation has almost invariant
vectors (Definition 3.12). It follows by Lemma 3.18 that λG has almost invariant vectors.
By (iv)⇒(i) of Theorem 3.19, we conclude that G is amenable.
3.8. Dye’s theorem and the Ornstein-Weiss theorem
The goal of this section is to establish the Ornstein-Weiss theorem (Theorem 3.79),
which states that any two ergodic p.m.p. actions of an amenable group on an atomless probability space are orbit equivalent. The Connes-Feldman-Weiss theorem (Theorem 3.68) and Example 3.54 reduce this problem to the case of Z, which was established
by Dye. Our efforts will thus be focused on proving Dye’s theorem (Theorem 3.78). The
strategy is to fix a single transformation and make repeated tandem use of Lemmas 3.72
and 3.73 to produce a nested sequence of ladders (Definition 3.74) which together model
a dyadic odometer in the limit. Since dyadic odometers are all conjugate, as observed
before Theorem 3.78, this yields Dye’s theorem.
Recall from Example 3.50 that two p.m.p. actions G y (X, µ) and H y (Y, ν) are
said to be orbit equivalent if there are conull sets X ′ ⊆ X and Y ′ ⊆ Y with GX ′ ⊆ X ′
and GY ′ ⊆ Y ′ and a measure isomorphism ϕ : X ′ → Y ′ such that ϕ(Gx) = Hϕ(x) for
every x ∈ X ′ . We also apply this terminology to single transformations by way of the
Z-action they generate.
Let T be a measure-preserving transformation of an atomless standard probability
space (X, µ). Note that if T is ergodic then it is automatically free, since for every n ∈
Z\{0} the set {x ∈ X : T n x = x} is Borel by Proposition A.21 and hence must have
measure zero by ergodicity and Theorem A.20. The full group [T ] is defined as the set of
3.8. DYE’S THEOREM AND THE ORNSTEIN-WEISS THEOREM
137
all measure-preserving transformations S : X → X such that for a.e. x ∈ X the S-orbit
{S n x : n ∈ Z} is contained in the T -orbit {T n x : n ∈ Z}. For the purposes of studying
[T ] as a topological group, which we do not do here, it should be more precisely defined as
a set of equivalence classes modulo a.e. equality, but in any case its elements are typically
expressed and manipulated as genuine transformations.
L EMMA 3.72 (Rokhlin lemma). Suppose that T is free. Let n ∈ N and ε > 0. Then
there is a Borel set A ⊆ X such that the sets T k A for k = 0, . . . , n − 1 are pairwise
S
k
disjoint and µ( n−1
k=0 T A) ≥ 1 − ε.
P ROOF. Let C be the collection of countable families P of a.e. pairwise disjoint T invariant Borel subsets of X such that for every Y ∈ P there is a Borel set A ⊆ Y
S
k
for which the sets T k A for k = 0, . . . , n − 1 are pairwise disjoint and µ( n−1
k=0 T A) ≥
(1 − ε)µ(Y ). We define an order on C by declaring that P P ′ if every member of P is
equal a.e. to a member of P ′ . Now if C′ is a totally ordered subcollection of C, then we
S
S
can find a sequence {Pn } in C′ such that supn∈N µ( Pn ) = supP ∈C′ µ( P ), in which
S
′
case ∞
n=1 Pn is an upper bound for C . Since C contains ∅ and hence is nonempty, it
follows by Zorn’s lemma that C has a maximal element P .
S
We now need only show that µ( P ) = 1, for we can then witness the statement of
the lemma by taking the union over Y ∈ P of sets A ⊆ Y as in the definition of C.
S
Suppose to the contrary that µ( P ) < 1. Pick an integer m > 1/ε. Since T is free,
S
by Proposition 1.4 we can find a nonnull set B ⊆ X\ P such that the sets T k B for
k = 0, . . . , mn − 1 are pairwise disjoint. By Poincaré recurrence (Theorem 1.10), B is
partitioned, modulo a null set, by the Borel sets
Bk = {x ∈ B : T k x ∈ B and T j x ∈
/ B for j = 1, . . . , k − 1}
F
F⌊k/n⌋−1 jn
for k ≥ mn. Set A = ∞
T Bk . Then the sets T k A for k = 0, . . . , n − 1
k=mn
j=0
are pairwise disjoint, and the complement of their union inside the T -invariant set Y0 :=
S∞
F∞ Fk−1
n
j
n=0 T B is equal to
k=mn
j=⌊k/n⌋n T Bk , which has measure at most
k−1
∞
G
X
n
j
·µ
T Bk ,
k
j=0
k=mn
S
which is at most µ(Y0 )/m and hence less than εµ(Y0 ). Since P is T -invariant, the
S
complement of P is T -invariant and therefore contains Y0 modulo a null set. By adding
S
Y0 to P we then contradict maximality. We conclude that µ( P ) = 1, as desired.
L EMMA 3.73. Suppose that T is ergodic. Let A and B be Borel subsets of X. Then
there is an S ∈ [T ] such that SA = B if and only if µ(A) = µ(B).
138
3. AMENABILITY
P ROOF. As the elements of [T ] are measure-preserving, the forward direction is immediate. For the converse, suppose that µ(A) = µ(B). Set à = A\B and B̃ = B\A.
Take an enumeration k1 , k2 , . . . of Z and recursively define for n ∈ N the pairwise disjoint
sets
n−1
n−1
[
[
−kn
km
An = Ã ∩ T
B̃\
\
T Am
Am .
m=1
m=1
S
S
Then ∞
An = Ã modulo a null set. Indeed if this is not the case then since Ã\ ∞
n=1
n=1 An
S
kn
and B̃\ ∞
T
A
have
the
same
measure
we
can
use
Proposition
1.5
to
find
a
nonnull
n
n=1
subset A′ of the first of these two sets that maps into the second under T kn for some n, in
S
km
which case T kn A′ ⊆ n−1
Am by the definition of An , a contradiction. Now as An
m=1 T
kn
and T An are disjoint for every n, we can define an S ∈ [T ] by setting S = T kn on An
and S = T −kn on T kn An for each n, and Sx = x for all x ∈ X\(Ã ∪ B̃). Then SA = B,
as desired.
n−1
D EFINITION 3.74. A ladder for T is a pair ({Ci }n−1
i=0 , S) where {Ci }i=0 is a Borel
partition of X and S is a measure isomorphism from C0 ⊔ · · · ⊔ Cn−2 to C1 ⊔ · · · ⊔ Cn−1
such that SCi = Ci+1 for i = 0, . . . , n − 2 and for a.e. x ∈ C0 ⊔ · · · ⊔ Cn−2 the point
Sx is contained in the T -orbit of x. We refer to each Ci as a rung and to n as the ladder’s
height.
Note that we can always extend an S as above to an element of [T ] by setting S =
(S n−1 )−1 on Cn−1 , which we will do in the proof of Lemma 3.77. This extension is
n-periodic, i.e., S n = id and S k x 6= x for all x ∈ X and k = 1, . . . , n − 1.
L EMMA 3.75. Suppose that T is ergodic. Let n ∈ N and ε > 0. Then there is a ladder
L = ({Ci }n−1
i=0 , S) with n rungs such that µ({x ∈ dom S : Sx 6= T x}) < ε.
P ROOF. By Lemma 3.72 there is a Borel set A ⊆ X such that the sets T i A for i =
S
k
0, . . . , n − 1 are pairwise disjoint and the complement B := X\ n−1
k=0 T A has measure
less than ε. Since µ is atomless, by Theorem A.20 we can find a partition {B0 , . . . , Bn−1 }
of B into n sets of equal measure, and by Lemma 3.73 we can find T0 , . . . , Tn−2 ∈ [T ]
such that Tj Bj = Bj+1 for every j = 0, . . . , n − 2. Then the ladder L = ({Ci }n−1
i=0 , S)
defined by setting
(i) Ci = Bi ∪ T i A for i = 0, . . . , n − 1,
S
i
(ii) S = T on n−2
i=0 T A and S = Tj on Bj for j = 0, . . . , n − 2
has the desired properties.
′
We say that a ladder L′ refines another ladder L = ({Ci }n−1
i=0 , S) if L has the form
S
k−1
kn−1
′
for i = 0, . . . , n − 1, and S ′ = S on
({Ci′ }i=0 , S ′ ) where k ∈ N, Ci = j=0 Ci+jn
Sn−2
i=0 Ci .
3.8. DYE’S THEOREM AND THE ORNSTEIN-WEISS THEOREM
139
L EMMA 3.76. Suppose that T is ergodic. Let L = ({Ci }n−1
i=0 , S) be a ladder, A
a Borel subset of X, and ε > 0. Then for every integer k > 2n /ε there is a ladder
′
′
L′ = ({Ci′ }kn−1
i=0 , S ) of height kn refining L such that there is a collection of rungs of L
′
′
whose union A satisfies µ(A∆A ) < ε.
P ROOF. Let {D1 , . . . , Dm } be the partition of C0 which generates the same algebra as
the sets S −i (A ∩ Ci ) for i = 0, . . . , n − 1, and note that m ≤ 2n . Let k be an integer larger
than 2n /ε, and for every j = 1, . . . , m take an integer qj ≥ 0 such that qj /kn ≤ µ(Dj ) <
(qj + 1)/kn. Since µ is atomless, we can use Theorem A.20 to find for each j = 1, . . . , m
a collection Cj of qj pairwise disjoint Borel subsets of Dj of measure 1/kn, and then a
partition {B0 , . . . , Bk−1 } of C0 containing C1 ∪· · ·∪Cm whose members all have measure
1/kn. By Lemma 3.73 there are T0 , . . . , Tk−2 ∈ [T ] such that Tj (S n−1 Bj ) = Bj+1 for
kn−1
every j = 0, . . . , k − 2. Now define the ladder L′ = ({Ci′ }i=0
, S ′ ) by setting
′
(i) Ci+jn
= S i Bj for i = 0, . . . , n − 1 and j = 0, . . . , k − 1,
S
′
n−1
(ii) S ′ = S on n−2
Bj for j = 0, . . . , k − 2.
i=0 Ci and S = Tj on S
For each i = 0, . . . , n − 1 write Ai for the union of all sets which have the form S i Q for
F
′
some Q ∈ C1 ∪ · · · ∪ Cm and are contained in A ∩ Ci . Set A′ = n−1
i=0 Ai . Then A is a
union of rungs of L′ which satisfies A′ ⊆ A and
2n
1
· mn ≤
< ε,
µ(A\A′ ) ≤
kn
k
so that A′ is as desired.
Given a ladder L = ({Cj }n−1
j=0 , S), every x ∈ X lies in a unique set of the form
2
n−1
{y, Sy, S y, . . . , S y} for some y ∈ C0 , and we define orbL (x) to be this set.
−1
L EMMA 3.77. Suppose that T is ergodic. Let L = ({Cj }N
j=0 , S) be a ladder and let
δ > 0. Then for every sufficiently large k there is a ladder L′ of height kN refining L
such that
µ({x ∈ X : T x ∈
/ orbL′ (x)}) < δ.
P ROOF. Write µ0 for the probability measure on C0 obtained by restricting µ and
scaling by µ(C0 )−1 . For a.e. x ∈ C0 there exists an n ∈ N such that T n (x) ∈ C0 , which
is a consequence of Poincaré recurrence (Theorem 1.10), and we write nx for the smallest
such n. Let T0 be the µ0 -preserving transformation of C0 defined by T0 (x) = T nx (x)
for a.e. x ∈ C0 (this is called the transformation induced by T on C0 ). By Lemma 3.75
and the comment immediately preceding it, there is a sequence {Tk }∞
k=1 of µ0 -preserving
transformations of C0 in [T0 ] such that Tk cyclically permutes the elements in a Borel
partition of C0 of cardinality k and
µ({x ∈ C0 : Tk x 6= T0 x}) → 0
140
3. AMENABILITY
as k → ∞. Let ψ : X → C0 be the map that collapses the levels of the ladder by way of
S, i.e., ψ = S −j on the level Cj . Then for every x ∈ X the points ψ(x) and ψ(T x) lie
in the same orbit of T and hence there is an n ∈ Z such that T0n ψ(x) = ψ(T x), since the
orbits of T0 are intersections of orbits of T with C0 . Note that for a fixed n the set of all
x ∈ X satisfying T0n ψ(x) = ψ(T x) is the union of the Borel sets {x ∈ Ci ∩ T −1 (Cj ) :
T0n S −i x = S −j T x} over i, j = 0, . . . N − 1 and hence is itself Borel. We can thus find
an n0 ∈ N and a large enough k ∈ N such that Tk x = T0 x for all x in a subset of C0 of
measure close enough to µ(C0 ) to guarantee that
(22)
µ({x ∈ X : Tkn ψ(x) = ψ(T x) for some n ∈ {−n0 , . . . , n0 }}) > 1 − δ.
By our choice of Tk , there exist Borel sets B0 , B1 , . . . , Bk−1 partitioning C0 such that
Tk Bi = Bi+1 for all i = 0, . . . , k − 2 and Tk Bk−1 = B0 . This defines a ladder for the
−1
′
transformation T0 , which we combine with L to produce a ladder L′ = ({Cj′ }kN
j=0 , S )
for T by setting
′
(i) Ci+lN
= S i Bl for i = 0, . . . , N − 1 and l = 0, . . . , k − 1,
S −2
′
−N +1
(ii) S ′ = S on N
on CN −1 \S N −1 Bk−1 .
i=0 Ci and S = Tk S
Then by (22) we have µ({x ∈ X : T x ∈
/ orbL′ (x)}) < δ.
A dyadic odometer is an odometer (see the fourth item of Section 1.3) for which the
Q
nk
product has the form ∞
− 1} for some nk ∈ N. All dyadic odometers are
k=1 {0, . . . , 2
Q
nk
conjugate, for we can define a conjugacy between the odometers on ∞
k=1 {0, . . . , 2 −1}
and {0, 1}N by using binary representations to identify {0, . . . , 2nk − 1} with the product
of the copies of {0, 1} over the indices from n1 + · · · + nk−1 + 1 to n1 + · · · + nk , and
then matching up the factors accordingly in the two products over N.
T HEOREM 3.78. Any two ergodic measure-preserving transformations of an atomless
probability space are orbit equivalent.
P ROOF. Let T be an ergodic measure-preserving transformation of an atomless probability space (X, µ). It suffices to show that T is orbit equivalent to a dyadic odometer, as
these are all conjugate by the discussion above.
Take a sequence {An } of Borel subsets of X which generates the σ-algebra such
that each set in the sequence appears infinitely often. Starting with the trivial ladder
2kn −1
L0 = ({X}, id), we recursively construct a sequence of ladders Ln = ({Cn,i }i=0
, Sn )
whose heights are powers of 2 such that Ln+1 refines Ln for every n ∈ N by applying
Lemma 3.76 and 3.77 in succession at a given stage n so that
(i) there is collection of rungs of Ln whose union A satisfies µ(An ∆A) < 1/n,
and
(ii) µ({x : T x ∈
/ orbLn (x)}) < 1/n.
3.9. NOTES AND REFERENCES
141
Note that whenever m > n the transformation Sm restricts to Sn on all rungs of Ln
apart from the topmost one, whose measure tends to zero as n → ∞. Let S be the
transformation in [T ] which for each n restricts to Sn on the complement of the top rung
of Ln . In view of (ii), the set B of all x ∈ X such that T x is contained in the S-orbit of
T
x has measure one. Then the set m∈Z T m B has measure one, and each point x in this
set has the property that its T -orbit is contained in its S-orbit, for if m > 0 then for some
n ∈ Z we have T m x = T (T m−1 x) = S n T m−1 x and by repeating this procedure m − 1
times we see that T m x is contained in the S-orbit of x, while if m < 0 then for some
n ∈ Z we have x = T −m (T m x) = S n T m x by the first case and so T m x = S −n x. In
view of the fact that S ∈ [T ], it follows that the set of all x ∈ X whose S- and T -orbits
are equal has measure one. Thus the identity map on X is an orbit equivalence between
S and T .
Finally, in view of (i) we see that the transformation S is measure-algebraically conQ
kn −kn−1
jugate to the odometer on ∞
− 1}, and hence conjugate to it by Then=1 {0, . . . , 2
orem 0.5.
T HEOREM 3.79. Any two ergodic p.m.p. actions G y (X, µ) and H y (Y, ν) of
countable amenable groups on atomless probability spaces are orbit equivalent.
P ROOF. By Example 3.54 and Theorem 3.68, every p.m.p. action of a countable
amenable group is orbit equivalent to a measure-preserving transformation. As ergodicity
for a p.m.p. action is a property of the orbit equivalence relation (since the G-invariance
of a Borel set clearly only depends on the orbit equivalence relation), we can now apply
Theorem 3.78 to obtain the result.
3.9. Notes and references
The concept of amenability was introduced by von Neumann in [140] in connection
with the Banach-Tarski paradox and related problems about invariant measures. General
references are [64, 114, 116, 18]. Amenability has become a staple not only in group
theory but also in areas of functional analysis involving multiplicative structure, such
as C∗ -algebras and von Neumann algebras, where it sets the framework for the Elliott
classification program and Connes’s classification of injective factors (see [38, 17, 130,
21]). One can even detect the spirit of amenability in the minimality/tightness dichotomy
theorem of [42] for Banach spaces. For a wealth of information on paradoxicality and its
relation to amenability and the existence of invariant measures, see [143].
In Theorem 3.4, the equivalence (ii)⇔(vi) is due to Tarski [132, 133] and the equivalence (i)⇔(v) to Følner [44]. Our proof of (vi)⇒(v) in Theorem 3.4 is taken from [19],
which was inspired by [28] and makes use of a matching theorem of Rado (see [102, 14]),
which we have absorbed into our presentation. Prior to [19] there did not exist a direct
142
3. AMENABILITY
argument for this Tarski-to-Følner implication that did not pass through invariant means
or measures. Indeed the Følner and Tarski characterizations of amenability for a discrete
group G typically appear separately and in different contexts. Tarski proved that nonparadoxicality implies the existence of a finitely additive invariant measure on G in two steps.
The first involves the construction of a semigroup (the “type semigroup”) from the finite
collections of subsets of G modulo the relation of equidecomposability, and then showing
by a Hahn-Banach-type argument that the absence of a state on this semigroup imples an
(n+1)-into-n version of nonparadoxicality. In the second step the latter is shown to imply
2-into-1 paradoxicality (in the sense of Definition 3.3(iii)) using the axiom of choice. The
standard proof that amenability implies the Følner property, which replaces the combinatorics of Følner’s original approach with functional-analytic techniques, passes through
Reiter’s property by concatenating arguments of Day [26] and Namioka [106]. One starts
by expressing an invariant mean on ℓ∞ (G) as a weak∗ limit of a net of nonnegative normone functions fi in ℓ1 (G), so that (sfi − fi ) → 0 weak∗ for all s ∈ G. Since these
differences lie in ℓ1 (G), one can view the convergence as occurring in the weak topology
on ℓ1 (G), and the same holds for sequences of tuples {(sfi − fi )}s∈F in ℓ1 (G)F where
F is any finite subset of G. By Mazur’s theorem, for any such F we can replace the
fi with suitable convex combinations to get norm convergence. By doing this across an
increasing sequence of finite subsets of G whose union is all of G, one can then construct
a sequence of functions witnessing Reiter’s property (Definition 3.3(ii)). Finally, using a
layer cake argument one winnows out an asymptotically invariant sequence of normalized
characteristic functions, whose supports are then the desired Følner sets.
For many years it was an open question, implicit in the work of von Neumann and
explicitly posed by Day, whether a nonamenable G must contain a copy of the free group
F2 . This was determined to be false in the late 1970s by Olshanskii, who showed that
for every large enough prime p there exists an infinite group whose nontrivial proper subgroups are all cyclic of order p, and, applying a criterion of Grigorchuk, that such groups
(called Tarski monsters) are nonamenable [111]. The group of piecewise projective homeomorphisms of the line is another example of a nonamenable group not containing F2 , as
recently shown by Monod [103].
The class of elementary amenable groups is the smallest class of groups which contains Abelian groups and finite groups and is closed under taking subrgoups, quotients,
extensions, and increasing unions. Answering a question of Day, Grigorchuk gave an
example of a finitely generated amenable group (known as the Grigorchuk group) which
is not elementary amenable. The obstruction to elementarity is that the growth of the
number of words in a fixed finite set of generators is neither polynomial nor exponential (“intermediate growth”). The Grigorchuk group is part of the rich class of groups
generated by automata [109].
3.9. NOTES AND REFERENCES
143
Topological full groups provide another source of examples. Let T : X → X be
a minimal homeomorphism of the Cantor set. Consider the topological full group JT K
consisting of all homeomorphisms S : X → X for which there is a clopen partition of
X such that on each member of the partition S restricts to T n for some n ∈ Z. Since
there are only countably many clopen partitions of X, this group is countable. Matsui
showed that the commutator subgroup of JT K is (i) simple, (ii) not finitely presented,
and (iii) finitely generated precisely when T is conjugate to the restriction of the shift on
{1, . . . , k}Z for some k ∈ N to some closed invariant set [99]. Monod and Juschenko
proved moreover in [75] that JT K is amenable, and hence that the commutator subgroup is
amenable (Proposition 3.2(ii)). This yielded the first examples of simple finitely generated
infinite amenable groups. If we replace Z by Z2 then amenability for the topological full
group of a minimal action is no longer guaranteed [37].
Section 3.2 is based on the article [7] by Bekka, who uses the term amenable to refer
to unitary representations satisfying Definition 3.11. Our proof of the Powers-Størmer
inequality, Lemma 3.13, is from [118]. Lemmas 3.14 and 3.15 are from [21].
The mean ergodic theorem in Section 3.3 was established for single transformations
by von Neumann [141] and in the more general amenable case by Dye [34]. Sections 3.5
and 3.6 are based on the influential article [112] by Ornstein and Weiss, whose motivation
was to develop the entropy theory of actions of amenable groups so as to be able to extend
Ornstein isomorphism theorem for Bernoulli shifts to this setting. For information on
monotilability see [145]. The pointwise ergodic theorem in Section 3.4 was established
by Birkhoff [12] for single transformations and by Lindenstrauss [94] in the amenable
case. We have given the proof of Ornstein and Weiss that appears in [146].
A classical source of information about p.m.p. equivalence relations is the article [41]
by Feldman and Moore. Zimmer introduced the concept of amenability into this context
in [150]. The Connes-Feldman-Weiss theorem (Theorem 3.68) is from [22]. We have
presented the original argument, which is operator-algebraic in spirit (compare [21]). A
more geometric approach which derives hyperfiniteness from Reiter’s property can be
found in [77]. See also [84]. We have followed [84] in our formulation of Lemma 3.55
as a purely Borel-theoretic result. Dye’s theorem (Theorem 3.78) appeared in [33] and
was fundamental in establishing the theory of orbit equivalence. Our exposition was influenced by [80]. The theme of measured equivalence relations (and measured groupoids
more generally) was also pursued by Mackey in [95] as the basis of his program on virtual
groups. The study of orbit equivalence is closely related to phenomena in von Neumann
algebras [104, 126, 53, 137].
In [113] Ozawa and Popa introduced the property of weak compactness for a p.m.p.
action G y (X, µ). This requires the existence of a net {Pn } of finite-rank projections
in B(L2 (X)) such that (i) kPn π(s) − π(s)Pn k2 → 0 for all s ∈ G, and (ii) kPn f −
144
3. AMENABILITY
R
f Pn k2 → 0 and Tr(Pn f ) → X f dµ for all f ∈ L∞ (X). One can then show using
the Connes-Feldman-Weiss theorem that the following are equivalent for a group G: (i)
G is amenable, (ii) every p.m.p. action of G is weakly compact, and (iii) every weakly
mixing p.m.p. action of G is weakly compact. Unlike compactness and weak mixing,
weak compactness is an orbit equivalence invariant.