Internat. J. Math. & Math. Sci.
Vol. 22, No. 4 (1999) 713–726
S 0161-17129922713-5
© Electronic Publishing House
SMOOTHNESS CONDITIONS ON MEASURES
USING WALLMAN SPACES
CHARLES TRAINA
(Received 15 October 1997 and in revised form 13 April 1998)
Abstract. In this paper, X denotes an arbitrary nonempty set, ᏸ a lattice of subsets of X
with ∅, X ∈ ᏸ, A(ᏸ) is the algebra generated by ᏸ and M(ᏸ) is the set of nontrivial, finite,
and finitely additive measures on A(ᏸ), and MR (ᏸ) is the set of elements of M(ᏸ) which
are ᏸ-regular. It is well known that any µ ∈ M(ᏸ) induces a finitely additive measure µ̄ on
an associated Wallman space. Whenever µ ∈ MR (ᏸ), µ̄ is countably additive.
We consider the general problem of given µ ∈ M(ᏸ), how do properties of µ̄ imply
smoothness properties of µ? For instance, what conditions on µ̄ are necessary and sufficient for µ to be σ -smooth on ᏸ, or strongly σ -smooth on ᏸ, or countably additive? We
consider in discussing these questions either of two associated Wallman spaces.
Keywords and phrases. Wallman spaces, outer measure associated with a lattice measure,
smoothness of a lattice measure, regular outer measure.
1991 Mathematics Subject Classification. 28A12, 28C15.
1. Introduction. We give some standard notational and background material in
Section 2, related matters can be found in [1, 4, 11]. In Section 3, we answer questions of smoothness of µ in terms of outer measures (associated with µ̄) of sets in the
Wallman spaces. Section 4 gives smoothness conditions for µ assuming the measurability of certain sets in the Wallman spaces.
Some related materials concerning these outer measures can be found in [5, 6, 8,
9, 10].
2. Background and notation. We introduce the necessary measure theoretic, and
lattice definitions. The definitions and notations are standard and are consistent with
those found in [1, 2, 3, 4, 11]. We collect the one’s we need for the reader’s convenience.
Throughout, X denotes an arbitrary nonempty set, ᏸ a lattice of subsets of X such
that ∅, X ∈ ᏸ. For E ⊂ X, E = X − E denotes its complement. We denote by A(ᏸ),
the algebra generated by ᏸ; δ(ᏸ), the lattice of all countable intersections of sets
from ᏸ; τ(ᏸ), the lattice of arbitrary intersections of sets from ᏸ; ᏸ , the lattice of
complements of sets from ᏸ. The necessary measure theoretic definitions follow.
The set of all nonnegative finite valued, nontrivial, finitely additive, and bounded
measures on A(ᏸ) is denoted by M(ᏸ).
An element µ ∈ M(ᏸ) is said to be σ -smooth on ᏸ if and only if whenever Ln ∈ ᏸ,
n = 1, 2, . . . , and Ln ↓ ∅, µ(Ln ) → 0.
An element µ ∈ M(ᏸ) is said to be σ -smooth on A(ᏸ) if and only if whenever An ∈
A(ᏸ), n = 1, 2, . . . , and An ↓ ∅, µ(An ) → 0. (We note that this condition is equivalent
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CHARLES TRAINA
to µ being countably additive.)
An element µ ∈ M(ᏸ) is said to be strongly σ -smooth on ᏸ if and only if whenever
∞
L, Ln ∈ ᏸ, n = 1, 2, . . . ; and Ln ↓ L, L = n=1 Ln , µ(L) = inf{µ(Ln ) | n = 1, 2, . . . }.
An element µ ∈ M(ᏸ) is said to be ᏸ-regular if and only if, for any A ∈ A(ᏸ),
µ(A) = sup{µ(L) | L ⊂ A, L ∈ ᏸ}.
The following notations is used to denote the subsets of M(ᏸ) determined by the
preceding properties
(a) Mσ (ᏸ) is the set of measures that are σ -smooth on ᏸ;
(b) M σ (ᏸ) is the set of measures that are σ -smooth on A(ᏸ);
(c) J(ᏸ) is the set of measures that are strongly σ -smooth on ᏸ;
(d) MR (ᏸ) is the set of ᏸ-regular measures;
(e) MRσ (ᏸ) is the set of ᏸ-regular measures of M σ (ᏸ).
We note that M σ (ᏸ) ⊂ Mσ (ᏸ), and MR (ᏸ) ∩ Mσ (ᏸ) ⊂ M σ (ᏸ). We denote by I(ᏸ),
Iσ (ᏸ), I σ (ᏸ), Φ(ᏸ), IR (ᏸ), and IRσ (ᏸ) the subsets of the corresponding M’s that consist
of the nontrivial 0–1 valued measures.
We write µ ≤ ν(ᏸ) whenever µ, ν are measures, or set functions such that µ(L) ≤
ν(L) for all L ∈ ᏸ.
We now recall some lattice definitions
(a) A lattice ᏸ is said to be T0 if and only if, given any two distinct points of X, there
is a lattice set containing one and not the other.
(b) A lattice ᏸ is said to be disjunctive if and only if, for any x ∈ X and A ∈ ᏸ such
that x ∈ A, there exists an L ∈ ᏸ such that x ∈ L, A ∩ L = ∅.
(c) A lattice ᏸ is normal if, for any A, B ∈ ᏸ with A∩B = ∅, there exists L, K ∈ ᏸ with
A ⊂ L , B ⊂ K , and L ∩ K = ∅.
(d) A lattice ᏸ is a delta lattice (δ-lattice) if δ(ᏸ) = ᏸ.
(e) A lattice ᏸ is compact if, for any collection {Lα } of sets from ᏸ, α Lα = ∅ implies
n
that there exists Lα1 , . . . , Lαn with i=1 Lαi = ∅.
(f) A lattice ᏸ is countably compact if and only if, for every sequence {Ln } from ᏸ
∞
k
with n=1 Ln = ∅, there exists Ln1 , . . . , Lnk with i=1 Lni = ∅.
(g) If ᏸ1 and ᏸ2 are two lattices of subsets of X, then ᏸ1 separates ᏸ2 if and only
if, whenever A, B ∈ ᏸ2 with A ∩ B = ∅ implies that there exists C, D ∈ ᏸ1 with A ⊂ C,
B ⊂ D, and C ∩ D = ∅.
For µ ∈ M(ᏸ), the support of µ is the set
S(µ) = L ∈ ᏸ | µ(L) = µ(X) .
(2.1)
We briefly recall two of the generalized Wallman spaces that will play a central role
in this paper. For further details, see [1, 2, 3, 11].
If ᏸ is T0 and disjunctive we define, for A ∈ A(ᏸ), W (A) = {µ ∈ IR (ᏸ) | µ(A) = 1}.
Clearly, for A, B ∈ A(ᏸ),
(1) W (A ∪ B) = W (A) ∪ W (B),
(2) W (A ∩ B) = W (A) ∩ W (B),
(3) W (A ) = W (A) ,
(4) A ⊂ B if and only if W (A) ⊂ W (B),
(5) W (A(ᏸ)) = A(W (ᏸ)).
SMOOTHNESS CONDITIONS ON MEASURES . . .
715
W (ᏸ) = {W (L) | L ∈ ᏸ} is taken as a base for the closed sets of a topology τW (ᏸ).
W (ᏸ) is a compact lattice and, therefore, τW (ᏸ) is compact and W (ᏸ) separates
τW (ᏸ). IR (ᏸ), τW (ᏸ) is a compact T1 space and is T2 if and only if ᏸ is normal. In
addition X is embedded in IR (ᏸ) by the mapping
X → IR (ᏸ),
x → µx ,
where µx is the Dirac measure concentrated at x ∈ X, i.e.,

1, if x ∈ A,
µx (A) =
A ∈ A(ᏸ).
0, if x ∈ A,
(2.2)
(2.3)
As known, specific choices of X and ᏸ yield well-known compactifications of X.
Similarly, if ᏸ is just T0 , we consider the second Wallman space: for A ∈ A(ᏸ), let
V (A) = {µ ∈ I(ᏸ) | µ(A) = 1}.
Properties (1)–(5) hold with V in place of W , and taking V (ᏸ) as a base for the closed
sets, we get the space I(ᏸ), τV (ᏸ) which is compact, T0 . Clearly, X is embedded in
I(ᏸ) by
X → I(ᏸ),
x → µx .
(2.4)
Next, let µ ∈ M(ᏸ) and define for A ∈ A(ᏸ),
µ̄ W (A) = µ(A).
(2.5)
It is easy to see that µ̄ ∈ M(W (ᏸ)), and that the mapping µ → µ̄ is a bijection from
M(ᏸ) to M(W (ᏸ)). Also, since W (ᏸ) is a compact space, M(W (ᏸ)) = Mσ (W (ᏸ)). If
µ ∈ MR (ᏸ), then µ̄ ∈ MR (W (ᏸ)) = MRσ (W (ᏸ)). Thus µ̄, in this case, can be extended
to σ (W (ᏸ)) even to Sµ̄∗ and is δW (ᏸ) regular on these sets. We recall the following
theorem from [2].
Theorem 2.1. Let ᏸ be a lattice of subsets of the nonempty set X, where ᏸ is T0 and
disjunctive. The following are equivalent
(1) µ ∈ MRσ (ᏸ);
∞
∞
(2) µ̄( i=1 W (Li )) = 0, i=1 W (Li ) ⊂ IR (ᏸ) − X, Li ↓, Li ∈ ᏸ;
∞
∞
(3) µ̄( i=1 W (Li )) = 0, i=1 W (Li ) ⊂ IR (ᏸ) − IRσ (ᏸ), Li ↓, Li ∈ ᏸ;
(4) µ̄ ∗ (X) = µ̄(IR (ᏸ)).
One of our main aims is to extend this theorem.
In a similar manner, for µ ∈ M(ᏸ) and A ∈ A(ᏸ), we define µ̄(V (A)) = µ(A). Again,
µ → µ̄ is a bijection of M(ᏸ) to M(V (ᏸ)) and M(V (ᏸ)) = Mσ (V (ᏸ)).
In either of the two Wallman spaces, certain smoothness properties of µ ∈ M(ᏸ) are
directly reflected to µ̄. For example, µ ∈ J(ᏸ)(M σ (ᏸ)) implies that µ̄ ∈ J(W (ᏸ))(M σ (W
(ᏸ))) and similarly for V (ᏸ). What we seek first in this paper is an analogue to
Theorem 2.1 without the regularity assumptions for smoothness. To elaborate, if
µ ∈ M(ᏸ), then µ̄ always belongs to Mσ (W (ᏸ))(Mσ (V (ᏸ))) and to obtain necessary
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CHARLES TRAINA
and sufficient conditions for a µ ∈ M(ᏸ) to be in Mσ (ᏸ) in terms of µ̄, we must look
elsewhere than just for smoothness on W (ᏸ)(V (ᏸ)).
We recall that, for a µ ∈ M(ᏸ), there are several associated outer measures (see
[6, 8, 9, 10]) which are defined as follows: for any E ⊂ X,
µ (E) = inf µ(L ) | E ⊂ L , L ∈ ᏸ ;
(2.6)
µ is just finitely subadditive;


∞
∞


µ (E) = inf
µ Li E ⊂
Li , Li ∈ ᏸ


i=1
(2.7)
i=1
and
µ ∗ (E) = inf

∞


i=1

∞

µ(Ai ) E ⊂
Ai , Ai ∈ A(ᏸ) .

(2.8)
i=1
The outer measures µ and µ ∗ are both countably subadditive.
Finally,
µ • (E) = inf µ(A) | E ⊂ A, A ∈ A(ᏸ) ,
(2.9)
which is only finitely subadditive.
We have the following relations between µ and these outer measures. µ • = µ on
A(ᏸ), µ ∗ ≤ µ on A(ᏸ). If, however, µ ∈ M σ (ᏸ), then µ ∗ = µ on A(ᏸ). It is clear that
µ = µ on ᏸ ; while µ ≤ µ on ᏸ and µ = µ on ᏸ if and only if µ ∈ MR (ᏸ). If µ ∈ Mσ (ᏸ),
then µ (X) = µ(X) and µ ≤ µ on ᏸ (see [6, 10]). There are other obvious relationships
that hold between these outer measures, however, we do not explicitly list them.
We next consider a finite outer measure ν defined on all subsets of X, which may be
either finitely or countably subadditive.
Sν = E ⊂ X | ν(G) = ν(G ∩ E) + (G ∩ E ) for all G ⊂ X
(2.10)
is the set of ν-measurable sets.
The outer measure ν is regular if, for every E ⊂ X, there exists an M ∈ Sν such that
E ⊂ M and ν(E) = ν(M). If ν is regular, then E ∈ Sν if and only if ν(X) = ν(E) + ν(E ).
If ν is a regular countably subadditive outer measure and if {En } is a sequence of sets
from X such that En ⊂ En+1 , n = 1, 2, . . . , then ν(limn→∞ En ) = limn→∞ ν(En ). (See [7]).
The following result is known (see [6, 10]).
Theorem 2.2. Let µ ∈ Mσ (ᏸ). Then µ = µ on ᏸ if
(a) ᏸ is countably compact;
(b) ᏸ is δ-normal;
(c) ᏸ is normal and countably paracompact.
Corollary 2.2.1. Let µ ∈ M(ᏸ), where ᏸ is T0 and disjunctive, and let µ̄ ∈
M(W (ᏸ)) = Mσ (W (ᏸ)) be the induced measure. Then µ̄ = µ̄ on W (ᏸ).
Remark. A similar result holds for µ ∈ M(ᏸ) with ᏸ just T0 , namely, if µ̄ ∈
M(V (ᏸ)) = Mσ (V (ᏸ)), then µ̄ = µ̄ on V (ᏸ).
SMOOTHNESS CONDITIONS ON MEASURES . . .
717
A number of these equalities can be extended. For instance, the following is known.
Theorem 2.3. Let ᏸ1 , ᏸ2 be lattices of subsets of X such that ᏸ1 ⊂ ᏸ2 and δ(ᏸ1 ) ⊂
ᏸ2 . Let µ ∈ Mσ (ᏸ1 ). Then if ᏸ1 separates ᏸ2 , µ = µ on ᏸ1 implies that µ = µ on ᏸ2 .
Corollary 2.3.1. If µ ∈ M(ᏸ) and if
(a) ᏸ is T0 and disjunctive, then µ̄ = µ̄ on τW (ᏸ).
(b) ᏸ is just T0 , then µ̄ = µ̄ on τV (ᏸ).
To make our statements concise, we note that in dealing with the Wallman space
IR (ᏸ), τW (ᏸ), we always assume that ᏸ is T0 and disjunctive; while in working with
the Wallman space I(ᏸ), τV (ᏸ), we always assume that ᏸ is T0 . In either case, for
E ⊂ X, we can view (via the mapping x → µx noted earlier) E embedded in IR (ᏸ) or
I(ᏸ).
We now note some relationships among µ , µ , and the induced outer measures
µ̄ , µ̄ .
Theorem 2.4. Let µ ∈ M(ᏸ) and let E ⊂ X. Then
(a) µ (E) = µ̄ (E),
(b) µ (E) = µ̄ (E),
(c) µ ∗ (E) = µ̄ ∗ (E),
(d) µ • (E) = µ̄ • (E),
(e) for L ∈ ᏸ,
µ (L) = µ̄ (L) ≤ µ̄ W (L) = µ̄ W (L) = µ (L),
µ (L) = µ̄ (L) ≤ µ̄ V (L) = µ̄ V (L) = µ (L).
(2.11)
In (a)–(e), µ̄ ∈ M(W (ᏸ)) or µ̄ ∈ M(V (ᏸ)) depending on which Wallman space we are
considering.
Finally, we note the following theorem:
Theorem 2.5. (a) µ ∈ MR (ᏸ) if and only if µ̄ ∈ MR (W (ᏸ)).
(b) µ ∈ J(ᏸ) implies that µ̄ ∈ J(W (ᏸ)).
(c) µ ∈ M σ (ᏸ) implies that µ̄ ∈ M σ (W (ᏸ)).
The corresponding results are true with V (ᏸ) in place of W (ᏸ).
3. Characterizations of measures in the Wallman spaces. In this section, we wish
to extend the basic Theorem 2.1 to measures in Mσ (ᏸ), J(ᏸ), and M σ (ᏸ). In other
words, we wish to characterize these measures in terms of their induced measures on
the two Wallman spaces that we have considered. The proofs for the two spaces are
virtually the same so we present only one of them. We begin with Mσ (ᏸ).
Theorem 3.1. Let ᏸ be a T0 and disjunctive lattice, and for µ ∈ M(ᏸ), µ̄ ∈ M(W (ᏸ))
denotes its induced measure in the Wallman space IR (ᏸ), τW (ᏸ).
(a) If µ ∈ Mσ (ᏸ), then µ̄ (X) = µ̄(IR (ᏸ)).
(b) If µ ∈ M(ᏸ) and if µ̄ (X) = µ̄(IR (ᏸ)) and if µ̄ is a regular outer measure, then
µ ∈ Mσ (ᏸ).
(c) If µ ∈ I(ᏸ), then µ ∈ Iσ (ᏸ) if and only if µ̄ (X) = µ̄(IR (ᏸ)).
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CHARLES TRAINA
Proof. (a) Let µ ∈ Mσ (ᏸ). Since W (ᏸ) is compact, µ̄ ∈ Mσ (W (ᏸ)). Then µ (X) =
µ̄ (X) by Theorem 2.4. Since µ ∈ Mσ (ᏸ), µ (X) = µ(X). The definition of µ̄ gives
µ̄ W (X) = µ(X), but W (X) = IR (ᏸ). Thus, µ̄ (X) = µ̄ IR (ᏸ) .
(b) Let µ ∈ M(ᏸ) so that µ̄ ∈ M(W (ᏸ)) = Mσ (W (ᏸ)) (since W (ᏸ) is compact). Let
∞
∞
Ln ∈ ᏸ, n = 1, 2, . . . , Ln ↓ ∅, n=1 Ln = ∅. Then in IR (ᏸ), W (Ln ) ↓ and n=1 W (Ln ) ⊂
IR (ᏸ) − X.
∞
Therefore, n=1 W (Ln ) ⊃ X in IR (ᏸ). By monotonicity of µ̄ , we have since X ⊂
∞
n=1 W (Ln ) ⊂ IR (ᏸ) that


∞

µ̄ (X) ≤ µ̄
W Ln  ≤ µ̄ IR (ᏸ) .
(3.1)
n=1
Since µ̄ ∈ Mσ (W (ᏸ)),
µ̄ IR (ᏸ) = µ̄ IR (ᏸ) .
(3.2)
By hypothesis, µ̄ (X) = µ̄(IR (ᏸ)). By (3.1) and (3.2),


∞

µ̄ IR (ᏸ) = µ̄
W Ln  .
(3.3)
Now, µ̄(IR (ᏸ)) = µ̄(W (X)) = µ(X). Since µ̄ is regular,


∞

µ̄
W Ln  = lim µ̄ W Ln .
(3.4)
n=1
n→∞
n=1
But
lim µ̄ W Ln ≤ lim µ̄ W Ln = lim µ̄ W Ln .
(3.5)
µ(X) ≤ lim µ̄ W Ln .
(3.6)
n→∞
n→∞
Hence,
However, µ̄ W Ln
n→∞
n→∞
= µ Ln for all n. So,
µ(X) ≤ lim µ Ln .
(3.7)
lim µ Ln ≤ µ(X).
(3.8)
n→∞
In X, Ln ↑ X, so
n→∞
It follows from (3.7) and (3.8) that limn→∞ µ(Ln ) = µ(X) or limn→∞ µ(Ln ) = 0. So, µ ∈
Mσ (ᏸ).
(c) This follows immediately since any zero-one valued outer measure is regular.
Remark 1. The theorem holds with ᏸ just T0 if we replace the Wallman space
IR (ᏸ), τW (ᏸ) with the space I(ᏸ), τV (ᏸ).
We can strengthen the previous theorem as follows.
Theorem 3.2. (a) Let ᏸ be T0 and disjunctive and let µ ∈ M(ᏸ).
(1) If µ ∈ Mσ (ᏸ), then µ̄ (IRσ (ᏸ)) = µ̄(IR (ᏸ)).
(2) If µ ∈ M(ᏸ) and if µ̄ (IRσ (ᏸ)) = µ̄(IR (ᏸ)) and if µ̄ is regular,then µ ∈ Mσ (ᏸ).
SMOOTHNESS CONDITIONS ON MEASURES . . .
719
(3) If µ ∈ I(ᏸ), then µ ∈ Iσ (ᏸ) if and only if µ̄ (IRσ (ᏸ)) = µ̄(IR (ᏸ)).
(b) Let ᏸ be T0 and µ ∈ M(ᏸ). Then
(4) If µ ∈ Mσ (ᏸ), then µ̄ (Iσ (ᏸ)) = µ̄ (I σ (ᏸ)) = µ̄(I(ᏸ)).
(5) If µ ∈ M(ᏸ), µ̄ (Iσ (ᏸ)) = µ̄(I(ᏸ)), and µ̄ is regular, then µ ∈ Mσ (ᏸ).
(6) If µ ∈ I(ᏸ), then µ ∈ Iσ (ᏸ) if and only if µ̄ (Iσ (ᏸ)) = µ̄(I(ᏸ)).
It is clearly just necessary to prove part of this theorem because of the similarity of
cases (a) and (b) and in light of the proof of the previous theorem. We, therefore, just
prove case (b), parts (4) and (5).
Proof. (4) We have X ⊂ Iσ (ᏸ) ⊂ I(ᏸ), where X is identified with its image in I(ᏸ).
By monotonicity of µ̄ ,
(3.9)
µ̄ (X) ≤ µ̄ Iσ (ᏸ) ≤ µ̄ I(ᏸ) .
Since µ ∈ Mσ (ᏸ), µ̄ (X) = µ̄(I(ᏸ)). Similarly, µ̄ ∈ Mσ (V (ᏸ)). So, µ̄(I(ᏸ)) = µ̄ (I(ᏸ)).
It follows from the preceding statements that µ̄ (Iσ (ᏸ)) = µ̄(I(ᏸ)). Similarly, X ⊂
I σ (ᏸ) ⊂ Iσ (ᏸ) ⊂ I(ᏸ) and so µ̄ (X) ≤ µ̄ (I σ (ᏸ)) ≤ µ̄ (Iσ (ᏸ)). By hypothesis, µ ∈
Mσ (ᏸ). So, µ̄ (X) = µ̄(I(ᏸ)) and by the preceding result, µ̄ (Iσ (ᏸ)) = µ̄(I(ᏸ)). It
follows from these results that µ̄ (I σ (ᏸ)) = µ̄(I(ᏸ)).
∞
(5) Let Ln ∈ ᏸ, n = 1, 2, . . . such that Ln ↓ ∅. Then in I(ᏸ), V (Ln ) ↓, and so n=1 V (Ln )
∞
⊂ I(ᏸ) − X. If ν ∈ n=1 V (Ln ), then ν(Ln ) = 1 for all n and so ν(Ln ) → 1. Hence,
∞
∞
ν ∈ Iσ (ᏸ). Therefore, n=1 V (Ln ) ⊂ I(ᏸ) − Iσ (ᏸ), and so Iσ (ᏸ) ⊂ n=1 V (Ln ). By
monotonicity of µ̄ , it follows from the preceding set inclusion that
∞
V Ln ≤ µ̄ I(ᏸ) .
(3.10)
µ̄ Iσ (ᏸ) ≤ µ̄
n=1
By hypothesis, µ̄ (Iσ (ᏸ)) = µ̄(I(ᏸ)). Since V (ᏸ) is compact, µ̄ ∈ Mσ (V (ᏸ)) and so
µ̄ (I(ᏸ)) = µ̄(I(ᏸ)). It follows from (3.10) and the preceding two equalities that
∞
V Ln .
µ̄ I(ᏸ) = µ̄ (3.11)
µ̄ I(ᏸ) = µ̄ V (X) = µ(X).
(3.12)
n=1
But
So,

µ(X) = µ̄

∞
n=1
Since µ̄ is regular,

µ̄ 
∞
n=1
For each n,

V Ln  .
(3.13)

V Ln  = lim µ̄ V Ln .
n→∞
µ̄ V Ln ≤ µ̄ V Ln = µ̄ V Ln = µ Ln .
(3.14)
(3.15)
By (3.13), (3.14), and (3.15), we get
µ(X) ≤ lim µ Ln .
n→∞
(3.16)
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CHARLES TRAINA
Since Ln ↓ ∅, Ln ↑ X, and so µ(Ln ) ≤ µ(X) for all n. Therefore,
lim µ Ln ≤ µ(X).
n→∞
(3.17)
It follows from (3.16) and (3.17) that µ(X) = limn→∞ µ(Ln ), whence, limn→∞ µ(Ln ) = 0
whenever Ln ↓ ∅ in ᏸ. Hence, µ ∈ Mσ (ᏸ).
We now turn our attention to the case µ ∈ M σ (ᏸ), and we wish to obtain analogs
to the previous two theorems. Again, we can consider either of the two pairs IR (ᏸ),
W (ᏸ) or I(ᏸ), V (ᏸ). We will consider the latter just for brevity.
Theorem 3.3. Let ᏸ be T0 and let µ ∈ M(ᏸ).
(1) If µ ∈ M σ (ᏸ), then µ̄ ∗ (X) = µ̄ ∗ (Iσ (ᏸ)) = µ̄ ∗ (I σ (ᏸ)) = µ̄(I(ᏸ)).
(2) If µ ∈ M(ᏸ) and if µ̄ ∗ is regular, then if either µ̄ ∗ (X) = µ̄(I(ᏸ)) or if µ̄ ∗ (I σ (ᏸ)) =
µ̄(I(ᏸ)), then µ ∈ M σ (ᏸ).
(3) If µ ∈ I(ᏸ), then µ ∈ I σ (ᏸ) if and only if µ̄ ∗ (X) = µ̄(I(ᏸ)), or µ̄ ∗ (I σ (ᏸ)) =
µ̄(I(ᏸ)).
Proof. The proof is similar to that of Theorem 3.2. We just prove (1) and part of (2).
(1) Let µ ∈ M σ (ᏸ). In I(ᏸ), X ⊂ I σ (ᏸ) ⊂ Iσ (ᏸ) ⊂ I(ᏸ). Since µ ∈ M σ (ᏸ), µ̄ ∈
M σ (V (ᏸ)). Thus, we have µ ∗ (X) = µ(X) and µ̄ ∗ I(ᏸ) = µ̄ I(ᏸ) . Then
µ̄ I(ᏸ) = µ̄ V (X) = µ(X) = µ ∗ (X) = µ̄ ∗ (X).
(3.18)
µ̄ ∗ (X) ≤ µ̄ ∗ I σ (ᏸ) ≤ µ̄ ∗ Iσ (ᏸ) ≤ µ̄ ∗ I(ᏸ) = µ̄ I(ᏸ) .
(3.19)
µ̄ ∗ (X) = µ̄ ∗ Iσ (ᏸ) = µ̄ ∗ I σ (ᏸ) = µ̄ I(ᏸ) .
(3.20)
But,
Therefore,
(2) Let µ ∈ M(ᏸ) with µ̄ ∗ regular, and µ̄ ∗ (I σ (ᏸ)) = µ̄(I(ᏸ)). Suppose that An ∈
∞
A(ᏸ), n = 1, 2, . . . with An ↓ ∅ so that n=1 An = ∅. Since An ↓ ∅ in X, V (An ) ↓ in
∞
∞
I(ᏸ) so that n=1 V (An ) ⊂ I(ᏸ) − X. If ν ∈ n=1 V (An ), then ν(An ) = 1 for all n
∞
and so ν(An ) → 1, whence, ν ∈ I σ (ᏸ). Therefore, n=1 V (An ) ⊂ I(ᏸ) − I σ (ᏸ), and so
∞
I σ (ᏸ) ⊂ n=1 V (An ). By monotonicity of µ̄ ∗ ,


∞
∗ σ
∗
V An  ≤ µ̄ ∗ I(ᏸ) ≤ µ̄ I(ᏸ) .
(3.21)
µ̄ I (ᏸ) ≤ µ̄
n=1
∗
σ
The hypothesis µ̄ (I (ᏸ)) = µ̄(I(ᏸ)), together with the preceding inequality, yields


∞
V An .
µ̄ I(ᏸ) = µ̄ ∗ 
(3.22)
n=1
Now the proof can be completed as in part (b)5 of Theorem 3.2.
Now, we turn our attention to the case µ ∈ J(ᏸ). To handle this case in a manner
similar to the cases of Mσ (ᏸ) and M σ (ᏸ), we must introduce the appropriate outer
SMOOTHNESS CONDITIONS ON MEASURES . . .
721
measures. We first note trivially that µ ∈ J(ᏸ) if and only if whenever Ln ↓ L, Ln , L ∈ ᏸ,
then µ(Ln ∩ L ) → 0. Next, for each L ∈ ᏸ, we consider the following outer measure.
Consider the covering class consisting of ∅ and ᏸ ∪ L. Now, define for E ⊂ X and
µ ∈ M(ᏸ),


∞
∞


µ Li ∪ L E ⊂
Li ∪ L , Li ∈ ᏸ .
(3.23)
µL (E) = inf


i=1
i=1
We note the following easily proved facts.
(1) µL is an outer measure for each L ∈ ᏸ.
= µ .
(2) µφ
(3) If µ ∈ I(ᏸ), then µ ≤ µL on ᏸ for any L ∈ ᏸ if and only if µ ∈ Φ(ᏸ).
(4) If µ ∈ M(ᏸ), then µL (X) = µ(X) for any L ∈ ᏸ.
(5) For E ⊂ X, µL (E) = µ̄W
(L) (E) and µL (E) = µ̄V (L) (E).
Proof. We just show (3) and (4).
(3) Suppose that µ ∈ Φ(ᏸ) and that there exists A ∈ ᏸ such that µ(A) = 1 and
µL (A) = 0 for some L ∈ ᏸ. Then
A⊂
∞
i=1
Li ∪ L
µ Li ∪ L = 0
and
for all i.
(3.24)
Therefore, µ(Li ∩ L ) = 1 for all i. So, µ(Li ) = 1 for all i, and µ(L ) = 1.
∞
∞
Now, n=1 (Li ∩ L ) ∩ A = ∅, so that n=1 Li ∩ A ⊂ L, and µ(Li ∩ A) = 1 for all i and
µ(L) = 0. This is a contradiction since µ ∈ Φ(ᏸ). (Namely, suppose that µ ∈ Φ(ᏸ) and
∞
∞
suppose that i=1 Ai ⊂ B with Ai , B ∈ ᏸ, and µ(Ai ) = 1 for all i. Then B ⊂ i=1 Ai ,
µ(Ai ) = 0. Therefore, µ (B ) = 0, and so µ(B ) = 0 since µ ∈ Φ(ᏸ). Therefore, µ(B) = 1.)
Conversely, suppose that µ ≤ µL for all L ∈ ᏸ. Then µ ≤ µφ
= µ and so µ ∈ Iσ (ᏸ).
Suppose that Ln ↓ L, Ln , L ∈ ᏸ and µ(Ln ) = 1 for all n and µ(L) = 0, so that µ(L ) = 1.
∞
Then L = n=1 Ln , µ(Ln ) = 0. We have X = L ∪ L = ( Ln ) L and µ(Ln ∪ L) = 0.
Therefore, µL (X) = 0, a contradiction, since 1 = µ(X) ≤ µL (X).
Therefore, we must have µ ∈ Φ(ᏸ).
(4) Clearly, µL (X) ≤ µ(X) for L ∈ ᏸ. Suppose that µL (X) ≤ µ(X) for some L ∈ ᏸ.
Then there exists Li ∈ ᏸ such that
X⊂
∞
i=1
Li ∪ L
and
∞
µ Li ∪ L < µ(X).
(3.25)
i=1
Therefore,


n
∞
n
µ Li ∪ L = lim
µ Li ∪ L ≥ lim µ 
Li ∪ L .
µ(X) >
i=1
But,
n
i=1 (Li ∪ L)
n→∞
n→∞
i=1
n
↑ X and so limn→∞ µ(
i=1 (Li ∪ L))
µL (X) = µ(X)
(3.26)
i=1
= µ(X), a contradiction. Hence,
for all L ∈ ᏸ.
(3.27)
722
CHARLES TRAINA
Once again, we have the option of considering the two Wallman pairs IR (ᏸ), W (ᏸ)
or I(ᏸ), V (ᏸ). We will just state and prove the result for the latter pair.
Theorem 3.4. Let ᏸ be T0 and µ ∈ M(ᏸ). Then
(1) If µ ∈ J(ᏸ), then
µ̄V(L) (X) = µ̄V(L) I σ (ᏸ) = µ̄V(L) Iσ (ᏸ) = µ̄ I(ᏸ) for all L ∈ ᏸ.
(3.28)
(2) If µ ∈ M(ᏸ) and if µ̄V(L) is regular for all L ∈ ᏸ, then if µ̄V(L) (X) = µ̄(I(ᏸ)) for all
L ∈ ᏸ, or if µ̄V(L) (Φ(ᏸ)) = µ̄(I(ᏸ)) for all L ∈ ᏸ or if µ̄V(L)) (I σ (ᏸ)) = µ̄(I(ᏸ)) for all
L ∈ ᏸ, then µ ∈ J(ᏸ).
(3) If µ ∈ I(ᏸ), then µ ∈ Φ(ᏸ) if and only if µ̄V(L) (X) = µ̄(I(ᏸ)) or µ̄V(L) (Φ(ᏸ)) =
µ̄(I(ᏸ)) or µ̄V(L) (I σ (ᏸ)) = µ̄(I(ᏸ)) for all L ∈ ᏸ.
Proof. We prove (1) and a part of (2).
(1) Let µ ∈ J(ᏸ). Then µ̄ ∈ J(V (ᏸ)). By properties (4) and (5) of the outer measure
µL , we have
µL (X) = µ̄V(L) (X) = µ(X)
for all L ∈ ᏸ.
(3.29)
Since
µ̄ I(ᏸ) = µ̄ V (X) = µ(X).
V (X) = I(ᏸ),
(3.30)
Now, X ⊂ I σ (ᏸ) ⊂ Iσ (ᏸ) ⊂ I(ᏸ) and so, by monotonicity of µ̄V(L) ,
µ̄V(L) (X) ≤ µ̄V(L) I σ (ᏸ) ≤ µ̄V(L) Iσ (ᏸ) ≤ µ̄V(L) I(ᏸ) .
(3.31)
By property (4), µ̄V(L) (I(ᏸ)) = µ̄(I(ᏸ)). It follows from this and inequality (3.31) that
µ̄ I(ᏸ) = µ̄V(L) I σ (ᏸ) = µ̄V(L) Iσ (ᏸ) = µ̄V(L) (X)
for any L ∈ ᏸ.
(3.32)
(2) Let µ ∈ M(ᏸ). We show that if µ̄V(L) is regular for all L ∈ ᏸ and if µ̄V(L) (X) =
µ̄(I(ᏸ)), then µ ∈ J(ᏸ).
Let Ln ∈ ᏸ, n = 1, 2, . . . , L ∈ ᏸ such that Ln ↓ L. Then Ln ∩ L ↓ ∅ and so in I(ᏸ),
∞
∞
V (Ln ∩L ) ↓. Hence, n=1 V (Ln ∩ L ) ⊂ I(ᏸ) − X, so that X ⊂ n=1 V (Ln ∪ L).
It follows, by monotonicity of µ̄V(L) , that

µ̄V(L) (X)
≤
µ̄V(L) 
∞
n=1
V

 ≤ µ̄ Ln ∪ L
V (L)
I(ᏸ) .
(3.33)
By hypothesis, µ̄V(L) (X) = µ̄(I(ᏸ)); and by definition, µ̄V(L) (I(ᏸ)) ≤ µ̄(I(ᏸ)). The
preceding argument shows that


∞
µ̄ I(ᏸ) = µ̄V(L) 
V Ln ∪ L  = lim µ̄V(L) V Ln ∪ L
(3.34)
n→∞
n=1
by regularity of µ̄V(L) . But, µ(X) = µ̄(I(ᏸ)) and we have
µ(X) = lim µ̄V(L) V Ln ∪ L .
n→∞
(3.35)
SMOOTHNESS CONDITIONS ON MEASURES . . .
723
For all n,
µ̄V(L) V Ln ∪ L ≤ µ̄ V Ln ∪ L = µ Ln ∪ L ≤ µ(X)
(3.36)
since Ln ∪L ↑ X. Hence, µ(X) = limn→∞ µ Ln ∪L . It follows from this that limn→∞ µ(Ln ∩
L ) = 0, whenever, Ln ↓ L in ᏸ. Therefore, µ ∈ M(ᏸ).
4. Measurability and smoothness. In this section, we wish to show how the measurability of certain sets in the Wallman spaces affect the smoothness of the given
measure in the original space. To be more specific, we first introduce the following
theorem.
Theorem 4.1. Let X be an arbitrary nonempty set and ᏸ a T0 lattice of subsets
containing ∅, X.
(a) Let µ ∈ M(ᏸ). Suppose that X ∈ Sµ̄ . Then X ∈ Sµ̄ and if µ̄ is regular, then
µ ∈ Mσ (ᏸ).
(b) Let µ ∈ M(ᏸ). If Iσ (ᏸ) ∈ Sµ̄ , then Iσ (ᏸ) ∈ Sµ̄ and if µ̄ is regular, then µ ∈
Mσ (ᏸ).
Proof. We prove (a) only since the proof of (b) is similar.
(a) Let µ ∈ M(ᏸ). Since V (ᏸ) is compact, µ̄ ∈ Mσ (V (ᏸ)). Since X ∈ Sµ̄ ,
µ̄ I(ᏸ) = µ̄ (X) + µ̄ I(ᏸ) − X .
(4.1)
Now, µ̄(I(ᏸ)) = µ(X) = µ̄ (X), and so the preceding equation gives µ̄ (I(ᏸ) − X) = 0.
Since µ̄ ≤ µ̄ , we have µ̄ (I(ᏸ) − X) = 0. Therefore, I(ᏸ) − X ∈ Sµ̄ and so X ∈ Sµ̄ .
And we must have µ̄ (X) = µ̄(I(ᏸ)).
In addition, if µ̄ is regular, then the preceding argument, together with Remark 1,
shows that µ ∈ Mσ (ᏸ).
Remark 2. Instead of considering the pair I(ᏸ), V (ᏸ) as in Theorem 4.1, we could
consider IR (ᏸ), W (ᏸ), assuming that ᏸ is T0 and disjunctive, to obtain statements
similar to (a) and (b) above. For example, in this case, if µ ∈ M(ᏸ) and if IRσ (ᏸ) ∈ Sµ̄
then IRσ (ᏸ) ∈ Sµ̄ and if µ̄ is regular, then µ ∈ Mσ (ᏸ).
We now investigate measurability and σ -smoothness on A(ᏸ), that is, with the assumption of countable additivity.
Theorem 4.2. Let X be an arbitrary nonempty set and ᏸ a T0 disjunctive lattice of
subsets of X such that ∅, X ∈ ᏸ. Let µ ∈ MR (ᏸ) (then µ̄ ∈ MRσ (W (ᏸ))).
(a) If X ∈ Sµ̄• , then X ∈ Sµ̄∗ and µ ∈ MRσ (ᏸ).
(b) If IRσ (ᏸ) ∈ Sµ̄• , then IRσ (ᏸ) ∈ Sµ̄∗ and µ ∈ MRσ (ᏸ).
Proof. The proofs are quite similar so we just prove (a).
(a) Let µ ∈ MR (ᏸ). Then µ̄ ∈ MR (W (ᏸ)). Since W (ᏸ) is countably compact, µ̄ ∈
Mσ (W (ᏸ)), whence, µ̄ ∈ MRσ (W (ᏸ)), and so µ̄ ∗ is regular.
Since X ∈ Sµ̄• ,
µ̄ • IR (ᏸ) = µ̄ • (X) + µ̄ • IR (ᏸ) − X .
(4.2)
724
CHARLES TRAINA
Now, µ̄ • (W (X)) = µ̄ • (IR (ᏸ)) and so µ̄ • (X) = µ̄ • (IR (ᏸ)). It follows from the preceding
argument that
(4.3)
µ̄ • (X) = µ̄ • (X) + µ̄ • IR (ᏸ) − X .
Therefore,
µ̄ • IR (ᏸ) − X = 0.
(4.4)
Since µ̄ ∗ ≤ µ̄ • , (4.4) shows that µ̄ ∗ (IR (ᏸ) − X) = 0. Therefore, IR (ᏸ) − X ∈ Sµ̄∗ and so
X ∈ Sµ̄∗ . Now, µ̄ ∈ M σ (W (ᏸ)). So, µ̄(IR (ᏸ)) = µ̄ ∗ (IR (ᏸ)).
Since X ∈ Sµ̄∗ ,
(4.5)
µ̄ ∗ IR (ᏸ) = µ̄ ∗ (X) + µ̄ ∗ IR (ᏸ) − X
and, by (4.4),
µ̄ ∗ IR (ᏸ) = µ̄ ∗ (X) = µ̄ IR (ᏸ) .
(4.6)
By (4.6) and Remark 2, µ ∈ M σ (ᏸ).
The hypothesis µ ∈ MR (ᏸ) shows that µ ∈ MRσ (ᏸ).
Remark 3. If, instead of the Wallman pair IR (ᏸ), W (ᏸ), we consider I(ᏸ), V (ᏸ)
with just the assumption that ᏸ is T0 , we get similarly
If µ ∈ MR (ᏸ), then
(a) If X ∈ Sµ̄• , then X ∈ Sµ̄∗ , and µ ∈ MRσ (ᏸ).
(b) If I σ (ᏸ) ∈ Sµ̄• , then I σ (ᏸ) ∈ Sµ̄∗ , and µ ∈ MRσ (ᏸ).
If we do not assume that µ ∈ MR (ᏸ) in either case but only that µ ∈ M(ᏸ), the problem of then determining in terms of measurable sets that µ ∈ M σ (ᏸ) becomes much
more difficult. This is because µ ∈ M(ᏸ) only implies that µ̄ ∈ Mσ (W (ᏸ)) or Mσ (V (ᏸ))
depending on the Wallman space considered, and not a priori, as in the regular case
that µ̄ is countably additive. We can, however, prove the following theorem.
Theorem 4.3. Let X be an arbitrary nonempty set and ᏸ a T0 lattice of subsets of
X such that ∅, X ∈ ᏸ. Let µ ∈ M(ᏸ). If I σ (ᏸ) ∈ Sµ̄ then I σ (ᏸ) ∈ Sµ̄∗ and µ ∈ M σ (ᏸ) if
µ̄ ∗ is regular and if µ̄ ∗ (I σ (ᏸ)) = µ̄ (I σ (ᏸ)).
Proof. Let µ ∈ M(ᏸ). Since V (ᏸ) is countably compact, µ̄ ∈ Mσ (V (ᏸ)), which
implies that µ̄ (I(ᏸ)) = µ̄(I(ᏸ)). We always have
(4.7)
µ̄ I(ᏸ) = µ̄ I(ᏸ) .
Since I σ (ᏸ) ∈ Sµ̄ ,
µ̄ I(ᏸ) = µ̄ I σ (ᏸ) + µ̄ I(ᏸ) − I σ (ᏸ) .
(4.8)
µ̄ I(ᏸ) = µ̄ I σ (ᏸ) + µ̄ I(ᏸ) − I σ (ᏸ) .
(4.9)
Therefore, by (4.7),
But
µ̄ I(ᏸ) = µ̄ V (X) = µ(X).
(4.10)
SMOOTHNESS CONDITIONS ON MEASURES . . .
725
Hence, by (4.9) and (4.10),
µ(X) = µ̄ I σ (ᏸ) + µ̄ I(ᏸ) − I σ (ᏸ) .
(4.11)
Now, µ(X) ≤ µ̄ (X) ≤ µ̄ (I σ (ᏸ)), and so by (4.10)
µ(X) ≥ µ(X) + µ̄ I(ᏸ) − I σ (ᏸ) .
(4.12)
So, 0 ≤ µ̄ (I(ᏸ) − I σ (ᏸ)) ≤ 0.
Therefore, µ̄ (I(ᏸ) − I σ (ᏸ)) = 0. Since µ̄ ∗ ≤ µ̄ ≤ µ̄ ,
µ ∗ I(ᏸ) − I σ (ᏸ) = 0.
(4.13)
Therefore, I(ᏸ) − I σ (ᏸ) ∈ Sµ̄∗ and so I σ (ᏸ) ∈ Sµ̄∗ .
It follows that
µ̄ ∗ I(ᏸ) = µ̄ ∗ I σ (ᏸ) + µ̄ ∗ I(ᏸ) − I σ (ᏸ) .
(4.14)
µ̄ ∗ I(ᏸ) = µ̄ ∗ I σ (ᏸ) .
(4.15)
µ̄ I(ᏸ) ≤ µ̄ I(ᏸ) ≤ µ̄ I(ᏸ) .
(4.16)
µ̄ I(ᏸ) = µ̄ I(ᏸ) .
(4.17)
Thus,
Now,
By (4.7),
We have µ̄ ∗ (I(ᏸ)) ≤ µ̄(I(ᏸ)) and so
µ̄ ∗ I(ᏸ) ≤ µ̄ I(ᏸ) .
(4.18)
µ̄ ∗ I(ᏸ) = µ̄ I(ᏸ) .
(4.19)
µ̄ I σ (ᏸ) = µ̄ ∗ I σ (ᏸ) = µ̄ ∗ I(ᏸ) ≤ µ̄ I(ᏸ) = µ̄ I(ᏸ) .
(4.20)
By (4.17),
By hypothesis and (4.15),
Therefore, the inequality shows that µ̄(I(ᏸ)) = µ̄ ∗ (I σ (ᏸ)). By Theorem 3.3(2), µ ∈
M σ (ᏸ).
Acknowledgement. The author wishes to thank the referees for several useful
suggestions made to enhance the clarity of the paper.
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8000 Utopia Parkway, Jamaica, NY 11439