A metric space associated with probability space

Internat. J. Math. & Math. Scl. VOL. 16 NO. 2 (1993) 277-282

277

A METRIC SPACE ASSOCIATED WITH A PROBABILITY SPACE

KEITHF. TAYLORandXIKUI WANG

Department

ofMathematics Universityof Saskatchewan

Saskatoon,Sask. CanadaSTN 0W0

(Received January 7, 1992)

ABSTRACT.

For

acomplete probabilityspace

(9t, I,P),

theset of allcomplete sub-(z-algebras

of )2,

S(Y:.),

is given a natural metric and studied. The questions of when

5’(Z)

is compact or connected are awswered and the important subset consisting of all continuous sub-a-algebras is shown tobe closed. Con,aections with Christensen’s metricon theyon

Neumann

_{subalgebrasofa}

Type

II]-factorarebriefly discussed.

KEY

WORDS

AND

PHRASES.

a-algebras,conditional expectations,metricspace,yon

Neu-mannalgebra.

1992 AMS SUBJECT CLASSIFICATION

CODES.

Primary"

28A05,

28A20. Secondary:

46L10.

1. INTRODUCTION.

Let

(f/, E,

P)

beacomplete probabilityspaceandlet

S(I)

denotethe set ofallcomplete sub-(z-algebrasofI]. Wewill defineametric don

5’(Z)

and investigatesometopological propertiesofthe

resultingmetricspace

(S(Z),d).

In

[1],

Boylanintroduceda metric

d’

on

3’(Z)

for thepurpose of studying convergence propertiesoftheconditional expectationoperatorsdefinedby varying sub-(z-algebrasofZ. Ourmetricturns out to beequivalent toBoylan’sand seemstobemore convenient forstudyusing fitnctionalanalytic methods.

Weprove, in section3,that

(S(Z),d)

iscompact if and only if

Z

ispurelyatomic. Wealso show that

(S(Y_,),d)

isconnected if andonlyif

Z

has at mostoneatom.

In

section4,weconsider the continuous sub-(z-algebrasof

Z

and show that they formaclosed nowheredense subsetof

(S(Z),

d).

Thereis acloseanalogy between probabilityspaces and von

Neumann

algebras witha faithful finitenormal trace.

In

fact,ourdefinitionof dismodeledon ametricdefinedbyChristensen

[2]

on the set of all yon Neumann subalgebrasofa

Type II-factor.

Christensen’s metrichasbeen useful

in the studyofthe index in

II-factors

(see

[3]

and

[5]).

Wegiveashort discussionofacommon

generalizationofourmetricand Christensen’s in section5.

2.

THE

METRICS.

In

thissection,wedefinethemetric dandBoylan’smetric

d’

and show thattheyareequivalent. If

Zo

S(Z),

then let

L(Zo)

L(f, Zo, P)

be consideredas aclosed subspaceof

L(f, Z,

P)

L

(o),

{f

G

L’’(0)

ilflloo

_<

1}.

ForEl,

E

ES(E),let

d(Vl, 2)

7.J-{

sll|) inf

Ilf

gll2,

sup inf

II.f

gll2}.

Itiseasytocheckthat, disa metricon

S().

Forthe metric

d’,

we use a definitiondueto

Rogge

[6]

which isaslight variationon Boylan

[1].

For

EI,E

ES(Z), let

d’(Ea,_E)

sup inf

P(AAB)V

sup inf

P(AAB).

AZBEE BEEA

d

(Boylan’s metric d" has Vreplacedby

+

andclearly

1/2d*

< d’ _<

d’,so and d* areessentially the

same).

Thearguments in

[1]

showthat

(S(E),d’)is

acompletemetricspace.

Forany

Eo

S(E),

let

0

denote theorthogonal projectionof

L(,Z,P)

onto

L(fl,

E0,

P),

considered as asubspace of

Lz(E).

Let

E

denote the rtriction of e to

L(E).

As is well

k.own, e and

E

are restrictions of the conditional expectation mapping of

L(fl, E,P)

onto

L(,

E0, P). We will use any of the well known special properties ofconditional expectation

withoutgiving references.

Wenownsider the relationship between themetricdand

d’.

THEOREM 1. Forany

E,

E

S(E),

d’(E,,

E)

d(E,,

)

22d’(,,

Z)(1

d’(E,,

E))

Thus

d’

and dareequivalent metrics.

PROOF. To prove the left. hand inequality, fix

A

E.

It is shown in 2.1 of

[4]

that

if

P(AAB)

isachieved at theset

C

{EE(XA)

>

},

where XA isthe indicator function of theset A.

Some

elementary manipulations show that

P(AC)

-

I-

Er(XA)

dP

Now

II-

XAII,

,

o

P(AC)

I1

x.ll,

-I

E(XA)II,

IIX*

E(*)ll,

IIX*

E(XA)II*

Thus

inf

P(AAB)

d(, ),

for all

A

.

Symmetric arguments apply for2,which gives

d’(,,

)

5

d(,,

,).

E

,

For

the right inequality, note that, for any

f

E

L(),

(f)

f

and

E*(f)

is the

(L

-norm)

closest element of

L(E)

to

f.

Thus,

sup inf

Ilf-gll

sup

I[f-E(f)ll,

g

sup

IIW’(f)-E(f)ll,

lL(hai() _{IL() fL(h}

Similarly for theother term in d(E,),sowehave the inequality

(,,2:)

p

IIE’(f)-S(f)ll

METRIC SPACE ASSOCIATED WITH A PROBABILITY SPACE 279

and theorem 3in

[6]

shows that

Thus,

lIE

-’

(f’)

E2(.f’)[[2

<_

v/2d’(E,,

E2)(1

he(E,,

Z;))

lIE"’

(.f)

E(/)II

_<

Thiscompletes theproofofthetheorem.

COROLLARY. Theidentity map isahomeomorphismof

(S(E),d’)

with

(S(E),d).

REMARK

1.

Not

onlyarcd and

d’

equivalentinthesensethat theydefine thesameopenscts inS(E), but thesamesequences areCauchy ineachof

(._q(E),d)

and

(S(E),d’).

Since

(S(),d’)

is

complete,wehave that(._q(S).d) isacomplete metricspace.

REMARK

2. The right hand inequality intheorem cannot be improved tooneoftheform d

_<

kd’

forsomepositiveconstant I,’,asisshownbythefollowing example.

EXAMPLE.

Let12

[0,

1],

with

P

denotingLebesguemeasureonZ,the Lebesgue measurable stbsets of

[0,

1].

For each a

>

0, let

E

denote the a-algebra generated by an atom

[0, a]

and

El"l(a,

1]

{B

-

S C _(a,

1]}.

Forany

A

,

let

B

Af’l(a,

1]

and

B

[O,

a]

U(

A

l"l(

a,

1]).

Then

B,B

fi

E,

and either

P(AAB)

or

P(AAB)

islessthan orequalto

.

If

A

[0,

],

then

P(AAB) >

,

forall

B

E,

Thus

d’(E,,

E)=

To

compute

d(E,Z),

note that, for any

f

L(E),

thefunction

f..(,,,]

is in

L(Y:.o)

and

]]f

f.Y(.]]l

-<

.

On theotherhand,if

f

_{.([0,]- .X(,,],}then

Eo(])

0almosteverywhere.

Thisimplies that

]].f-

gll

->

ll.f[]2

v/.

forany g(

L(E,).

Thus

d(Zo, E)

yrs.

With

E

E,

and

E

,

the inequalities in theorem become

_<

_<

2-()

(_<

’).

Thusnok existswith

d(Z,, E)

_<

kd’(,E)

for alla

(0,1).

3.

THE MAIN

RESULTS.

Weconsiderthemetric space

(S(), d)

andestablishthe_{following two theorems.} Since

(S(),

d’)

ishomeomorphicto

(S(F_,),d),

these characterizationsholdfor eithermetric.

THEOREM

2.

(S(F_,),d)

iscompact ifand only if

:E

ispurely atomic.

THEOREM 3.

(S(F.,),d)

isconnectedifand only if has atmost oneatom.

Before proving these theorems,weneed thefollowinglemma whichisafolkloreresult ofmeasure theory andcan beproven withaneasy applicationof

Zorn’s

lemma.

LEMMA

If

(X,.T’,u)

is a finite measurespace with no atoms, then there existsa map

F"

[0,1]

.T"

such that

F(s)

C_

F(t)

and

,(F(t))

tI(X)

forall s,t

.

[0,1]

withs

_<

t.

Wewill usethe notation, forA

E

and

Ao

{AB"

B

c= 0}

PROOF

OF

THEOREM

_2.

Assume

_that

(12, E,P)

_{is not}

completely atomic.

Let fie

and

12

be the continuous and atomic parts of fl respectively.

We

are _assuming

P(fl) >

_0.

Let

F

[0,1]

E

_{be such that}

F(I)

_{C_}

12,P(F(t))

280 K.F. TAYLOR AND X. NANG

A,,

U{

F[(k

)/2"-’,

{2k

I)/2"]

k 1, :2 ,2

"-foreachn 1,2,... Then P(A,,)

P(f/c)/2,

for eachn and

P(A,AA,,)

P(f/c)/2,

forn

-

m. Let

E,

debotethe a-algebra

generated

by

A,,

f]

E

and theatom

f\A,,

for eachn.

Let

f

denote the indicatorfunction of

F/\A,.

Then

f

E L(E,,) and forany9 5

L(Em),

ifm n,then₉differs from

f

byat_{least 7}oneither

(F/\A,,)

CI

A,or

(f\A,)f’l(fl\A,,).

Eachofthesesetshas probability

P(flc)/4.

Thus

Ill

91122

_>

p(flc)/16,

whichimplies

d(E,

E,,,)

_>

/P(fc)/4,

ifn

-fl

m.

Hence,

the sequence

(E,,),,__

has nocluster points and

S(E)

isnotcompact.

Conversely, suppose

(f,E,P)

is completelyatomic. Since

(S(.,),d)

iscomplete, compactness will follow ifwe show that

S(E)

istotally bounded. If

E

hasonly afinitenumber of atoms, then

S(E)

is afinite set and obviously compact. So suppose

{A,

A2

aredisjointatomsgenerating

E. For

>

0, chooseNsuch that

:__N+

P(A,,)

<

.

There

areonlyafinitenumberof

algebras

of

setscontained in the

algebra generated

by

{A

A2v

}.

Let

.A,...,

,4 denote these

algebras. For

<

j

<

k,let denotethea-algebra generated by

,4j

and

{AN+I,

AN+,.

..}.

Let

A

For

any

’

S(E) A

Iq

’

.Ai

forsomej,

<

j

_<

k. Then

’

C_

E

andit is easy tocheck that

d(E;,

Zj)

<

c. Therefore,

S(Z)

iscoveredbythe finite setof-balls centered at

,

<

j

<

k. Since

c

>

0wasarbitrary,

S(E)

istotally bounded.

PROOF OF THEOREM 3. Suppose Aand

B

aretwo distinct atoms in

.

Let

S’

denote the set of all

E0

in

S(E)

which haveanatomcontainingboth

A

and

B,

let

S

denotethe set ofall

0

in

S(Z)

whichhave two disjoint atoms containing

A

and

B,

respectively. Itiseasy toseethat

S(E)

S

[J

S.

Let

S

and

E

S.

Let

A’

and

B’

be disjoint atoms of

E

containing

A

and

B,

respectively. Let Cbe an atom of

El

which contains

AI.JB.

Let

f

XA’

X’

-

L()

For

anyg

L(Z),g

a,almost everywhereon

C,

forsomeconstant a,

lal

<

1. Thus

IIf

gll

>

fA

If

gl

de

+/

If

gl

de

/(

-a)P(a)

+

(

+

a)P(B)

>

/2min{P(a),P(B)}

Therefore,

d(,E) >

/min{P(A),P(B)},

for all

r

S

and

Z

E

S.

Noticealso that the trivialo-algebra

E

{f,

1}

S

and

E

S.

So

S

and

S

arenonempty disjointopensubsets of

S(E)

with union

S(E). Hence,

S(E)

is notconnected if therearetwoor moreatomsin ).?,.

Conversely,suppose

(f/, E, P)

has at mostoneatom. If thereis anatom, callit

A

and if there isnoatom, let

A

.

If

P(A)

1, then

S(E)

isaonepoint space, whichisconnected.

So

assume

P(Ft\A)

>

0. Wewillshow that any

E0

S(E)

canbe connected toE; byanarcin

S(E).

Firstassume

E0

S(E)

and

E0

iscompletelyatomic,generated bythe disjoint atomsA0,

A,...

(a

finiteor infinite collection). We may assume

A

C_

Ao.

Foreach n

_>

1, by the lemma, there exists

F,,

[0,1]

A,,f’IE such that

P(F,(t))

tP(A)

and

F(s)

C_

F,(t)if

s,t

[0,1]

with s

<

t. Similarly, let

Fo

[0,1]

A01DE

be such that

F0

is increasing and

P(Fo(t))

P(A)

+

tP(Ao\a),

for all

[0, 1].

Foreach

[0,1],

define

Z(t)

S’()

as ther-algebra generated by

{F,(t)CIE, A,,\F,(t)’n

0,1,2

}.

It

isclear that

E(0)

Eo,

E(1)

E

and

E(s)

C_

Z(t)if

METRIC SPACE ASSOCIATED WITH A PROBABILITY SPACE 281

For

any

.[

6_.

L((t))l

and for each

,

0,1,2,...,

f

isconstant,saya,,,almost everywhereon the atom

.4,,\F,,(t)i,,

E(/). Define g E

L’’(E()),,

by making 9 a,, on

A,,\F,,(s)

and9

f

on

Fn(s),

forn 0, 1,2 Then

f

andy differonlyon

U(F,,(t)\F,,(s))

and therebyatmost2.

Now

P{(F,,(t)\F,(s)))

(t

s)[P(Ao\A)

+

P(A,)

+

P(A2)

+...]

Th

I1- 11

<

This implies that

d(V(s),V(t))

<

2

//-’Z-

s. Therefore,any atomic element

:E0

of

S()

is

con-nected byan arcto v

If

:E0

is not completely atomic, let f

tic

(J fl, where

flc["l

0

is continuous and

fl

:E0

is atomic.

Let

:E

be theatomic a-algebra generated by an atomequal to

fie

and

fl

]0.

By

methodsimilar tothe above, but justworking with fie,an arcfrom

:El

to

:E0

canbefound. Also,

beingatomicis connected byan arc to

.

Thusanyelement

:E0

of

S(:E)

can be connectedby an arcto :E. Therefore

S()

isconnected, in fact,arcwise connected if

(fl, :E,

P)

has atmostone atom.

The two theorems ofthis sectionshow that thereis somemeaningflfl connection between the

topologyof

S(:E)

and thestructure of theoriginal probabilityspace. Since this

topology

isreallyan

embodimentof the equiconvergence property for conditional expectations,wefeel that

S(:E)

with this metricd

(or

ifoneprefers

d’)

will provideagoodlocale forstudyingconditionalexpectations. 4.

CONTIIIUOUS SUB-a-ALGEBRAS.

Let

S(:E)=

{0

6

S(:E)’

(fl:E0,

P)has

no

atoms},

the space ofcontinuous sub-a-algebras

of E. Ofcourse, if is not acontinuous a-algebra, then

S()

.

On

the other

hand,

if

:E

is continuous, then

S(:E)

isavery rich set tolookat. For example,if

G

isagroup of :E-measurable transformations of12, let denote thea-algebraof G-invariant sets in

.

IfGis afinitegroup, then

e S(:E).

This providesaway in whichtoconstructmanyinteresting examplesofelements of

S(:E).

In

spite ofthis,

Sc(:E)

isa"small" subset of

S(:E).

THEOREM

4.

S(E)

isaclosed nowhere dense subset of

S(]).

PROOF. We

firstobserve that if

:E0

E

S(:E)

and

A

E,

then

{P(AB)"

B

e :Eo}

[0,

P(A)].

Nowfix

e S()\S(:E)

and letAbeanatomof

.

By

the aboveobservation,forany

:Eo

e

there exists

B

e

:Eo

such that

P(A["IB)

1/2P(A).

Let

f

_{X -Xu\}

L(0).

Forany g

L(F,),g

is constanton

A

andoneeasily checks that

Ill-

gll2

>-(P(A)) /:.

Thus

d(:E0, :E) _> (P(A))

/_forany

₀

e

S(:E).

_Hence

S()\S(:E)

_{isopen and}

S(:E)is

_closed.

To

see that

S(])

is nowhere dense, let

:E0

E

Sc()

and e

>

0. Choose

A

forthis particular

E0

and1,

d(

x"

,-,o,)

(P(A))

/ _{Thusthe open}

ballof radiusecenteredat

E0

is not contained in

S.(E)

forany

>

0.Since

5’c(E)

is

closed,

it isnowhere densein

S().

Thus

(So(E),

d)

is itselfaconplete metric space _{whose topological} propertiesshould be

inter-esting to study.

We

donot consider suchastudyinthis paper.

5. CONNECTIONS WITH VON

NEUMANN ALGEBRAS.

There is aone-.to-onecorrespondence between the sub-a-algebrasof

E

and the yonNeumann subalgebrasof

L(2, E,

P).

In

this section,weassumethe readeris familar with the basictheory

ofvon

Neumann

algebrasasfound ina reference such asSa"kai

[7].

Let

.M

beafixedfiniteyon

Neumann

algebrawithadistinguished faithful, finite, normaltrace

r. For example,

.M

could

be

L(,E,P)

or a

Type

lI-factor.

Let

S(.JM)

denote thesetofallyon Neumann subalgebrasof21 whichhave thesameidentityas.A4.

In

the case that

.M

is a

Type

ll-factor, Christensen defined a metric on

S(.M)

in

[2].

He showed that

S(AJ)

then becameacompletemetricspace.

He

used hismetrictostudy perturbation

properties of subfactors and his metrichas also been used in thestudyof the index ofasubfactor in a

Type

II-factor

(see

[3]

and

[5]).

Christensen’s definition works well in thesituation we are considering. Forx .A4,let

IIllz

9r(’)

For

A/,

S(.A4),

let

A/’

and denote theunit balls inA/’and

,

respectively.

Then define

d(A/’,

)

max{

sup inf x

YlI,

sup inf

II

x

YlI}

It

is routinetocheck thatd isametricon

S(.A4)

and Christensen’sproof that

S(.A4)

iscomplete

carriesovertothis moregeneralsituation.

Someoftheproofsofsection3canbeadaptedtotheyon

Neumann

_algebra situationbut not all.

We

list below what we can prove and leave the details of the adaption of the proofs tothe

interestedreader.

THEOREM 5. Let .A4 be a finite yon Neumann algebra with a fixed faithful, finite trace

r. With the Christensen’smetric,

S(.M)

iscompact if and onlyif.A4

i.s

generated byitsminimal projections.

PROPOSITION6. If

S(./t4)

isconnected,then.A4 hasat mostoneminimalprojection.

OPEN PROBLEM Istheconverseof proposition 6

true?

ACKNOWLEDGEMENT.Thisresearchwassupported byan

NSERC

Canada operating grant.

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Ann_.

_{Math._.__.}

Stst.

42

(1971),

552-559. 2. Christensen, E. SubalgebrasofaFiniteAlgebra, Math____:.

Ann___..

243 (1979),

17-29. 3.

Jones,

V. F.

1.Index for

Subfactors, Invent.

Math.

(1983),

_1-25.

4. Kudo, H.

A Note

onthe

Strong Convergence

ofr-algebras, Ann____. _2

(1974),

"/6-83. 5.

Mashhood, B.

and Taylor,K. F. On Continuityof theIndexofSubfactorsofa Finite

Factor,

J.

Func.

Anal. 12

(195S),

56-66.

6.

Rogge,

L.

Uniform Inequalities for_{Conditional Expectations,}

Ann;

_{Prob.___..2}

(1974),

486-489.