AND WORDS

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 the^yon

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 ofallcompletesub-(z-

algebrasofI]. Wewill defineametric don

5’(Z)

and investigatesometopological propertiesofthe resulting^metricspace

(S(Z),d).

In

[1],

^{Boylanintroduceda metric}

d’

on

3’(Z)

^{for thepurpose of}

studying convergence properties^oftheconditional 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 form^aclosed 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)

in thenatural way. Since bounded functions^aresquareintergrableonaprobabilityspace,wemay consider

L(Z0)

^{as a}

(non-closed)

^{subspace ofthe} Hilbert space

L(f, Z,P).

Itis easy to check that the unit ball in

L(Z0)

^{is closed in}

L(f, Z, P).

^{Themetric dis} essentially the Hausdorff metriconthe unit balls in theL2-norm. Let

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 isa}slight variationon Boylan

[1].

For

EI,E

ES(Z), let

d’(Ea,_E)

sup inf

P(AAB)V

sup inf

P(AAB).

AZBEE BEEA

(Boylan’s metric d" has Vreplacedby

+

^andclearly

1/2d* < d’ _<

d’,sod and d* areessentially the

same).

^Thearguments in

[1]

^showthat

(S(E),d’)is

^acomplete^metricspace.

Forany

Eo S(E),

^let

0

denote theorthogonal projection^of

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(h^ai() IL() fL(h

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

(,,2:)

p

IIE’(f)-S(f)ll

However,

for any

.f

^L’() the function

f’ ⁽¹ + f)

takes valuesin

[0, 1]

^almosteverywhere

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 thesame}sequences 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 in^theorem 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)) tP(fl),

and

F(s)

C_

F(t)

for s,t

. ^{[0,1],s _<}

^t.

Define

F[s,t] _F(t)\F(s)

and let

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 forany}9 5

L(Em),

ifm n,then₉differs from

f

^byatleast 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 no}cluster 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 haveanatomcontaining^both

A

and

B,

let

S

^{denotethe set of}all

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 to^E; 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 the}r-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

s_<t. Fixs,

t[0,1]withs<t.

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 the}structure 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 of}this,

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 letAbe}anatomof

. ^By

^{the above}observation,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

" ^:Eo

^{such that}

0<

P(A)

<^e

.

^Define

^:E

^tobethea-algebra generatedby

A

and

{B :E0" B

C

fl\A}.

^Then

has anatomA. Forany

f L(:E0),

let g

Xu\Af e L(:E).

^Since

Ill- ^gll] f.a Ill <- ^P(A)

and

:E

C:E0,we have that

d(0, E) _< (P(A))

^/

<

e. Combined with thegeneralestimateabove,

forthis particular

E0

^and1,

d(

^x"

,-,o,) (P(A))

^/ Thus_{the 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

beafixedfinite^yon

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 subalgebras^of21 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.

Someoftheproofsofsection3canbeadaptedtothe^yon

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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1. Boylan, E. S. EquiconvergenceofMartingales,

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 forConditional Expectations,

Ann;

Prob.___..2

(1974),

486-489.

7. Sakai,S.

C’-algebras and W’-algebr,

Ergebnisse derMath. 60, Berlin, Heidelley, New_York:

Springer-Verlag

1971.