The fundamental theorem of arithmetic for metric measures
spaces
Steven N. Evans
Department of Mathematics & Department of Statistics
Group in Computational and Genomic Biology
Group in Computational Science and Engineering
University of California at Berkeley
September, 2014
Steven N. Evans
Metric measure spaces
Collaborator
Ilya Molchanov
Bern
Steven N. Evans
Metric measure spaces
Cartesian product
Figure: The Cartesian product of two graphs.
Steven N. Evans
Metric measure spaces
Cartesian product
Formally, the Cartesian product GlH of two graphs G and H with vertex
sets V pGq and V pH q and edge sets E pGq and E pH q is the graph with
vertex set V pGlH q : V pGq V pH q and edge set
E pGlH q : tppg 1 , hq, pg 2 , hqq : pg 1 , g 2 q P E pGq, h P V pH qu
Y tppg, h1 q, pg, h2 qq : g P V pGq, ph1 , h2 q P E pH qu.
This operation is commutative and associative with the trivial graph as
identity element if we treat isomorphic graphs as being equal.
Steven N. Evans
Metric measure spaces
Cartesian product
A nontrivial graph is irreducible if it is not the Cartesian product of two
nontrivial graphs.
Sabidussi (1960) showed that any finite graph is a Cartesian product of
irreducible graphs and the factorization is unique up to order.
Factoring graphs and, more generally, embedding them in Cartesian
products is widely studied in computer science following Graham and
Winkler (1984, 1985).
Steven N. Evans
Metric measure spaces
Shortest path metric
Figure: A shortest path between two points in the Cartesian product of two graphs.
Steven N. Evans
Metric measure spaces
Shortest path metric
If two connected finite graphs G and H are equipped with the shortest
path metrics rG and rH , then the shortest path metric on the Cartesian
product is given by rGH rG ` rH , where
prG ` rH qppg1 , h1 q, pg2 , h2 qq : rG pg1 , g2 q rH ph1 , h2 q,
pg1 , h1 q, pg2 , h2 q P G H.
What happens if we extend this binary operation to more general metric
spaces?
Steven N. Evans
Metric measure spaces
Digression: Ulam’s problem
Ulam constructed a metric on the Cartesian product
of two metric spaces
a
pY, rY q and pZ, rZ q via ppy1 , z1 q, py2 , z2 qq ÞÑ rY py1 , y2 q2 rZ pz1 , z2 q2
and asked: Is it possible that there could be two non-isometric metric
spaces U and V such that the metrics spaces U U and V V are
isometric?
An example of two such spaces was given by Fournier (1971).
Such an example is not possible if U and V are compact subsets of
Euclidean space – Gruber (1971) & Moszyńska (1990).
Steven N. Evans
Metric measure spaces
Digression: Cancellativity
Ulam’s problem is closely related to the question of cancellativity for this
binary operation: If Y Z 1 and Y Z 2 are isometric, then are Z 1 and Z 2
isometric?
Cancellativity does not hold in general; for example, `2 pNq `2 pNq and
`2 pNq are isometric, but `2 pNq and the trivial metric space are not
isometric.
Cancellativity does not even hold for arbitrary subsets of R – Herburt
(1994).
There are compact Hausdorff topological spaces K with the property that
if L1 and L2 are two compact Hausdorff spaces such that K L1 and
K L2 are homeomorphic, then L1 and L2 are homeomorphic – Behrends
and Pelant (1995).
Steven N. Evans
Metric measure spaces
Back to “our” binary operation
Recall that we combine two metric spaces pY, rY q and pZ, rZ q into the
metric space pY Z, rY ` rZ q.
If a compact metric space is isometric to a product of finitely many
compact irreducible metric spaces, then this factorization is unique up to
the order of the factors – Tardif (1992).
Proof uses results on median algebras, Chebyshev sets, gated spaces from
Isbell (1980), Helíková (1983), Dress & Scharlau (1987).
There are certainly compact metric spaces that are not isometric to a
finite product of finitely many irreducible compact metric spaces.
Are there (unique) factorizations using some sort of infinite product?
What does this mean?
The study of this binary operation seems to be generally rather difficult.
Steven N. Evans
Metric measure spaces
Metric measure spaces
A metric measure space is just a complete separable metric space pX, rX q
equipped with a probability measure µX that has full support.
Two such spaces are equivalent if they are isometric as metric spaces via
an isometry that maps the probability measure on the first space to the
probability measure on the second.
Denote by M the set of such equivalence classes.
We do not distinguish between an equivalence class X
representative triple pX, rX , µX q.
Steven N. Evans
Metric measure spaces
P M and a
When are two metric measures spaces equivalent?
Gromov and Vershik showed that a metric measure space pX, rX , µX q is
uniquely determined by the probability distribution of the infinite random
matrix of distances
pdpξi , ξj qqpi,jqPNN ,
where pξk qkPN is an i.i.d. sample of points in X with common probability
distribution µX .
This concise condition for equivalence makes metric measure spaces
considerably easier to study than complete separable metric spaces per se.
Steven N. Evans
Metric measure spaces
A binary operation
Given two elements Y pY, rY , µY q and Z
be X pX, rX , µX q P M, where
`
p ` qpp q p
qq rY py1 , y2 q
p qp
qP b
X: Y
Z,
rX : rY
rZ , where
rY
rZ y 1 , z 1 , y 2 , z 2
y1 , z1 , y2 , z2
Y
Z),
µX : µY
µZ .
pZ, rZ , µZ q of M, let Y ` Z
rZ z 1 , z 2 for
p
q
This binary operation is associative and commutative.
The isometry class of metric measure spaces E that each consist of a
single point with the only possible metric and probability measure on them
is the identity element.
Thus pM, `q is a commutative semigroup with an identity (i.e. a monoid).
Steven N. Evans
Metric measure spaces
To paraphrase T.W. Körner
“The ease with which we proved [the central limit] explains why Fourier analysis
plays a rôle in probability theory that in other branches of mathematics is
played by thought.”
Steven N. Evans
Metric measure spaces
Semicharacters
A semicharacter is a map χ : M Ñ r0, 1s such that χpY ` Z q χpY qχpZ q
for all Y, Z P M.
Denote by A the family of arrays of the form A paij q1¤i j ¤n
n P N.
For each A P A define a semicharacter χA by
χA ppX, rX , µX qq :
Two elements X , Y
A P A.
»
exp
Xn
¸
¤ ¤
aij rX pxi , xj q
P Rp q for
n
2
n
µb
X pdxq.
1 i j n
P M are equal if and only if χA pX q χA pY q for all
Steven N. Evans
Metric measure spaces
Putting a metric on M
Equip M with the Gromov-Prohorov metric of Greven, Pfaffelhuber &
Winter (2009). Two elements of M are close if their random distance
matrices are close in distribution.
The space pM, dGPr q is complete and separable.
dGPr pX1 ` X2 , Y1 ` Y2 q ¤ dGPr pX1 , Y1 q
pX , Y q ÞÑ X ` Y is continuous.
limnÑ8 Xn
X
dGPr pX2 , Y2 q and so
if and only if limnÑ8 χA pXn q χA pX q for all A P A.
Steven N. Evans
Metric measure spaces
Putting a partial order on M
Define a partial order
some X P M.
For any Z
¤ on M by declaring that Y ¤ Z if Z Y ` X
P M, the set tY P M : Y ¤ Z u is compact.
Steven N. Evans
Metric measure spaces
for
Cancellativity
The commutative semigroup pM, `q is cancellative; that is, if
X , Y, Z 1 , Z 2 P M satisfy
X Y ` Z1
and
Y ` Z 2,
X
then
Z1
Steven N. Evans
Z 2.
Metric measure spaces
Measuring the size of a metric measure space
Put DA pX q : log χA pX q and
DpX q : D1 pX q log χ1 pX q log
Put
»
X2
2
exp prX px1 , x2 qq µb
X pdxq.
»
prX px1 , x2 q ^ 1q µbX2 pdxq.
For suitable constants, aDpX q ¤ DA pX q ¤ bDpX q.
a
1
RpX q ¤ dGPr pX , E q ¤ RpX q.
4
For suitable constants, αpDpX q ^ 1q ¤ RpX q ¤ β pDpX q ^ 1q.
R p X q :
X2
Steven N. Evans
Metric measure spaces
Convergence of sums
Suppose that pXn qnPN is a sequence such that limnÑ8 X0 ` ` Xn Y
for some Y P M. If pXn1 qnPN is a sequence that is obtained by re-ordering
the sequence pXn qnPN , then limnÑ8 X01 ` ` Xn1 Y also.
The limit limn°
Ñ8 X0 ` ` Xn exists if and only if
(equivalently, n RpXn q 8).
Ð
°
n
D p Xn q 8
So, we can make sense of sPS Xs for any family pX qsPS without
specifying the “order of summation”.
Steven N. Evans
Metric measure spaces
Irreducible elements
An element X P M is irreducible if X
that Y is either E or X .
E and Y ¤ X
for Y
P M implies
It is not clear a priori that there are irreducible elements. For example, the
semigroup R with the usual addition operation no irreducible elements in
the analogous sense.
If X P MztE u, then there is an irreducible element Y
this seems to be not at all obvious.
P M with Y ¤ X
Also, the irreducible elements are a dense Gδ subset of M.
Steven N. Evans
Metric measure spaces
–
Prime elements
An element X P M is prime if X
implies that X ¤ Y or X ¤ Z.
E and X ¤ Y ` Z for Y, Z P M
Prime elements are clearly irreducible, but the converse is not a priori true.
There are commutative, cancellative semigroups where the analogue of the
converse is false.
All irreducible elements of M are prime.
The analogous result for the integers is the key to proving the fundamental
theorem of arithmetic and the usual proof uses Euclid’s algorithm.
Steven N. Evans
Metric measure spaces
Prime factorization – the “fundamental theorem of arithmetic”
Given any X P MztE u, there is either a finite sequence pXn qN
n0 or an
infinite sequence pXn q8
n0 of irreducible elements of M such that
X X0 ` ` XN in the first case and X limnÑ8 X0 ` ` Xn in
the second.
The sequence is unique up to the order of its terms.
Each irreducible element appears a finite number of times, so the
representation is specified by the irreducible elements that appear and
their finite multiplicities.
It follows that pM, ¤q is a distributive lattice: there is an analogue of the
greatest common divisor (meet) and the least common multiple (join).
Steven N. Evans
Metric measure spaces
No nonconstant nondecreasing continuous functions
If Φ : R
Ñ M is a continuous function such that Φp0q E and
Φpsq ¤ Φptq, 0 ¤ s ¤ t 8,
then Φ E.
If Φ : R
Ñ M is a function such that Φp0q E and
Φps tq Φpsq ` Φptq, 0 ¤ s, t 8,
then Φ E.
Steven N. Evans
Metric measure spaces
Infinitely divisible random elements – Lévy-Hin̆cin-Itô
A random element Y of M is infinitely divisible if for each positive integer
n there are i.i.d. random elements Yn1 , . . . , Ynn such that Y has the
same distribution as Yn1 ` ` Ynn .
An infinitely divisible random element Y has the same distribution as
ð
tX : pt, X q P Πu,
where Π is a Poisson point process on r0, 1s pMztE uq with intensity of
the form λ b ν, where λ is Lebesgue measure and and ν is a σ-finite
measure on MztE u such that
»
DpX q ^ 1 ν pdX q 8.
Conversely, any such measure ν corresponds to an infinitely divisible
random element in this way.
Constants are not infinitely divisible and there is no analogue of the
Gaussian distribution.
Steven N. Evans
Metric measure spaces
Scaling
Given X
P M and a ¡ 0, set aX : pX, arX , µX q P M.
This scaling operation operation is continuous and satisfies
apX
` Y q paX q ` paY q.
If
paX q ` pbX q cX
for some X P M and a, b, c ¡ 0, then X E, so the second distributivity
law certainly does not hold.
Steven N. Evans
Metric measure spaces
Stable random elements – “LePage” representations
A M-valued random element Y is stable with index α ¡ 0 if for any
a, b ¡ 0 the random element
pa
bq α Y
1
has the same distribution as
a α Y1 ` b α Y2 .
1
1
A stable random element is necessarily infinitely divisible.
The index must satisfy 0 α 1.
An α-stable random element has the same distribution as
ð
P
1
Γn α Zn ,
n N
where pΓn qnPN is the sequence of successive arrivals of a homogeneous
unit intensity Poisson point process on R and pZn qnPN is a sequence of
i.i.d. random elements of M.
Steven N. Evans
Metric measure spaces
Future directions
Cancellativity allows us to embed the semigroup M into a group –
analogous to passing from N to Z.
Are there analogues of objects such as Gaussian random variables in this
world?
Rieffel has shown that one can obtain a “quantum” analogue of the space
of compact metric spaces equipped with the Gromov–Hausdorff distance
by considering C -algebras with properties that generalize those of the
algebra of Lipschitz functions on a compact metric space. Is there a similar
“quantization” for metric measure spaces? Ongoing work with Benson Au.
Steven N. Evans
Metric measure spaces