A probabilistic ergodic decomposition result

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www.imstat.org/aihp 2009, Vol. 45, No. 4, 932–942

DOI: 10.1214/08-AIHP302

A probabilistic ergodic decomposition result

Albert Raugi

IRMAR, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France E-mail: raugi@univ-rennes1.fr

Received 14 April 2008; accepted 1 September 2008

Abstract. Let(X,X, μ)be a standard probability space. We say that a sub-σ-algebraBofXdecomposesμin an ergodic way if any regular conditional probability^BP with respect toBandμsatisfies, forμ-almost everyx∈X,∀B∈B,^BP (x, B)∈ {0,1}. In this case the equalityμ(·)=

XBP (x,·)μ(dx), gives us an integral decomposition in “B-ergodic” components.

For any sub-σ-algebraBofX, we denote byBthe smallest sub-σ-algebra ofXcontainingBand the collection of all setsA inXsatisfyingμ(A)=0. We say thatBisμ-complete ifB=B.

Let{Bi: i∈I}be a non-empty family of sub-σ-algebras which decomposeμin an ergodic way. Suppose that, for any finite subsetJofI,

i∈JB_i=

i∈JB_i; this assumption is satisfied in particular when theσ-algebrasB_i,i∈I, areμ-complete. Then we prove that the sub-σ-algebra

i∈IB_idecomposesμin an ergodic way.

Résumé. Soit(X,X, μ)un espace probabilisé standard. Nous disons qu’une sous-tribuBdeXdécompose ergodiquementμsi toute probabilité conditionnelle régulière^BPrelativement àBetμ, vérifie, pourμ-presque toutx∈X,∀B∈B,^BP (x, B)∈ {0,1}.

Dans ce cas l’égalitéμ(·)=

XBP (x,·)μ(dx), nous donne une décomposition intégrale en composantes “B-ergodiques.”

Pour toute sous-tribuBdeX, nous notonsBla plus petite sous-tribu deXcontenantBet tous les sous-ensembles mesurables deXdeμ-mesure nulle. Nous disons que la tribuBestμ-complète siB=B.

Soit{B_i:i∈I}une famille non vide de sous-tribus deXdécomposant ergodiquementμ. Supposons que, pour toute partie finie JdeI,

i∈JB_i=

i∈JB_i; cette hypothèse est satisfaite si les tribusB_i,i∈I, sontμ-complètes. Alors la sous-tribu i∈IB_i décompose ergodiquementμ.

MSC:28A50; 28D05; 60A10

Keywords:Regular conditional probability; Disintegration of probability; Quasi-invariant measures; Ergodic decomposition

1. Introduction

There are several versions of ergodic decomposition theorems in the literature (cf. [3,4,6–8]) which give an integral decomposition of a probability measureμ, on a standard measurable space, in ergodic components. Most of these decompositions are based on abstract results like Choquet’s theorem. A probabilistic approach which can prove to be more convenient due to the properties of the conditional expectation (see [2]) is the following. Let(X,X, μ)be a standard Borel probability space. LetBbe a sub-σ-algebra ofX. We denote by^BP a regular conditional probability ofBandμ. We say thatBdecomposesμin an ergodic way if forμ-almost everyx∈X,∀B∈B,^BP (x, B)∈ {0,1}.

In this case, the equalityμ(dx)=

X

BP (x,·)μ(dx)gives us an integral decomposition inB-ergodiccomponents.

In [7,8] Shimomura proves that the intersection of a decreasing sequence of separable sub-σ-algebras ofXdecom- posesμin an ergodic way. He also gives an example of a standard probability space and a suitable sub-σ-algebra for which the above decomposition is not ergodic.

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Let{B_i: i∈I}be a non-empty family of sub-σ-algebras which decomposeμin an ergodic way. Suppose that, for any finite subsetJ ofI,

i∈JB_i=

i∈JB_i. The aim of this paper is to prove that the sub-σ-algebra

i∈IB_i decomposesμin an ergodic way.

2. Preliminaries

It is not necessary to work with standard Borel spaces. We only need probability space for which any sub-σ-algebra has regular conditional probabilities. In this section we recall some results about this property. On a standard Borel space(X,X), it is well known that, for any probability measureμand any sub-σ-algebraBofX, there exists a regular conditional probability with respect toμandB.

Definition 2.1. Aσ-algebra on a setXis called separable if it is generated by a countable sub-algebra.

Proposition 2.2. Let(X,X)be a measurable space with a separableσ-algebraX.Then two positiveσ-finite mea- sures are equal if they coincide on a countable algebra generatingX.

Definition 2.3. A classCof subsets ofXis said to be compact if,for any sequence(C_n)_n_∈Nof elements ofCwith an empty intersection

n∈NC_n,there exists a natural integerpsuch thatp

n=0C_n=∅.

Definition 2.4. Let (X,X, μ) be a probability space. Let C be a compact subclass of X. We say that C is μ- approximating if

∀A∈X μ(A)=sup

μ(C):C∈C, C⊂A .

Definition 2.5. Let(X,X, μ)be a probability space.LetBbe a sub-σ-algebra ofX.We call regular conditional probability with respect toBandμa mapP fromX×Xto[0,1]such that:

(i) for anyx∈X,P (x,·)is a probability measure onX.

(ii) for anyA∈X,the mapx∈X→P (x, A)is a version of the conditional expectationEμ[1A|B];that is,this map isB-measurable and,for anyB∈B,

X

1A(x)1B(x)μ(dx)=

X

P (x, A)1B(x)μ(dx).

Then for any non-negative(or bounded)X-measurable functionf,the functionPf defined byPf (x)=

Xf (y)× P (x,dy)(expectation offwith respect to the probabilityP (x,·))is a version of the conditional expectationEμ[f|B].

Theorem 2.6 ([5], corollaire Proposition V-4-4). Let(X,X, μ)be a probability space with a separableσ-algebra Xcontaining aμ-approximating compact class.

Then,for any sub-σ-algebraBofXthere exists a regular conditional probability with respect toBandμ.

Remarks. Let(X,X, μ)be a probability space with a separableσ-algebraXcontaining aμ-approximating compact class.LetBbe a sub-σ-algebra ofX.

1. IfP andQare two regular conditional probabilities with respect toBandμthen,forμ-almost everyx∈X, the probability measuresP (x,·)andQ(x,·)are equal.

2. IfP is a regular conditional probability with respect toBandμ,then for anyB∈B,we have,forμ-almost everyx∈X,

P (x, B)=E_μ[1B|B](x)=1B(x)=δ_x(B)∈ {0,1}, whereδ_xis the Dirac measure at the pointx.

When theσ-algebraBis separable,from Proposition2.2we can permute “for anyB∈B” and “forμ-almost everyx∈X.”

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3. LetBbe the smallest sub-σ-algebra containingBand the collection of all setsAinXsatisfyingμ(A)=0.

One sees easily that any regular conditional probability with respect toBandμis a regular conditional probability with respect toBandμ.For two sub-σ-algebrasB₁andB₂ ofX,the sub-σ-algebraB₁∩B₂ is not necessarily equal toB₁∩B₂;consequentlyL²(X,B₁∩B₂, μ)is not necessarily equal toL²(X,B₁∩B₂, μ).

3. Main results

Throughout this section, we assume that(X,X, μ)is a probability space with a separableσ-algebraXcontaining a μ-approximating compact class. The preceding Remark 2 leads us to introduce the following definition.

Definition 3.1. We say that a sub-σ-algebraBofXdecomposesμin an ergodic way if one(and thus all)regular conditional probability^BP with respect toBandμsatisfies,forμ-almost everyx∈X,∀B∈B,^BP (x, B)∈ {0,1}.

From the preceding Remarks 2 and 3 it follows that:

• Any separable sub-σ-algebra ofXdecomposesμin an ergodic way.

• IfBdecomposesμin an ergodic way then so doesB. (Any regular conditional probability^BP with respect toB andμis a regular conditional probability with respect toBandμ. If Bdecomposesμin an ergodic way then, forμ-almost everyx∈X,^BP (x, B)∈ {0,1}for anyB∈Band a fortiori for anyB∈B. Which proves thatB decomposesμin an ergodic way.)

Lemma 3.2. LetBbe a sub-σ-algebra ofX,and let^BP be a regular conditional probability with respect toBandμ.

Then,forμ-almost everyx∈X,we have the probability equalities

BP (y,·)=^BP (x,·) for^BP (x,·)-almost everyy∈X and consequently for anyB∈B,we have forμ-almost everyx∈X,

∀A∈X

X

1A(y)1B(y)^BP (x,dy)=

X

BP (y, A)1B(y)^BP (x,dy).

The following proposition tells us that the sub-σ-algebraBofXdecomposesμin an ergodic way if and only if, in the last equalities, we can permute “for anyB∈B” and “forμ-almost everyx∈X.”

Proposition 3.3. LetBbe a sub-σ-algebra ofX,and let^BP be a regular conditional probability with respect toB andμ.

Then the two following assertions are equivalent:

(i) Bdecomposesμin an ergodic way;

(ii) Forμ-almost everyx∈X,^BP is a regular conditional probability with respect toBand^BP (x,·).

In this case,for any sub-σ-algebraCofBand any regular conditional probability^CP with respect toCandμ,for μ-almost everyx∈X,^BP is a regular conditional probability with respect toBand^CP (x,·);that is,forμ-almost everyx∈X,

∀(A, B)∈X×B

X

BP (y, A)1B(y)^CP (x,dy)=

X

1A(y)1B(y)^CP (x, dy).

Moreover,this assertion is true for any sub-σ-algebraCofXsuch that,forμ-almost everyx∈X,

CP^BP (x,·)^def=

X

CP (x,dy)^BP (y,·)=^CP (x,·).

This last property is satisfied whenCis a sub-σ-algebra ofB.

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Theorem 3.4. Let (X,X, μ)be a probability space with a separable σ-algebraX containing aμ-approximating compact class.Let{Bi: i∈I}be a non-empty family of sub-σ-algebras which decomposeμin an ergodic way.We suppose that,for any finite subsetJ ofI,

i∈JBi=

i∈JBi. Then the sub-σ-algebra

i∈IB_i decomposesμin an ergodic way.

4. Proof of the results

Throughout this section, we assume that(X,X, μ)is a probability space with a separableσ-algebraXcontaining a μ-approximating compact class. For any sub-σ-algebraBofX, we denote by^BP a regular conditional probability with respect toBandμ.

4.1. Proof of Lemma3.2

LetA∈X. The functionsg(x)=^BP (x, A)and(g(x))²areB-measurable. Therefore, forμ-almost everyx∈X,

BP g(x)=Eμ[g|B](x)=g(x) and ^BP g²(x)=g²(x)=_B P g(x)2

.

From the Cauchy–Schwarz equality it follows that, forμ-almost everyx∈X,

BP (x, A)=g(x)=g(y)=^BP (y, A) for^BP (x,·)-almost everyy∈X.

The first assertion of the lemma is then a consequence of Proposition 2.2.

For anyB∈Band forμ-almost everyy∈X,

BP (y, B)=Eμ[1B|B](y)=1B(y).

Asμ(dy)=

X

BP (x,dy)μ(dx), forμ-almost everyx∈X,

1B(y)=^BP (y, B) for^BP (x,·)-almost everyy∈X.

Hence, for anyA∈Xand forμ-almost everyx∈X,

X

1A(y)1B(y)^BP (x,dy)=

X

1A(y)^BP (y, B)^BP (x,dy)

=

X

1A(y)^BP (x, B)^BP (x,dy) (first assertion)

=^BP (x, A)^BP (x, B)

=

X

BP (x, A)1B(y)^BP (x,dy)

=

X

BP (y, A)1B(y)^BP (x,dy) (first assertion). (1)

The Proposition 2.2 allows us to permute “for anyA∈X” and “forμ-almost everyx∈X.”

4.2. Proof of Proposition3.3

LetX0be a measurable subset ofXsuch thatμ(X0)=1 and for anyx∈X0,

BP (y,·)=^BP (x,·) for^BP (x,·)-almost everyy∈X.

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(i)⇒(ii) If theσ-algebraBdecomposesμin an ergodic way, then there exists a measurable subsetX₁ofXsuch thatμ(X₁)=1 and for anyx∈X₁,

∀B∈B ^BP (x, B)∈ {0,1}.

Forx∈X₀∩X₁, we have for any(A, B)∈X×B,

X

1A(y)1B(y)^BP (x,dy)=^BP (x, A)^BP (x, B)=

X

BP (y, A)1B(y)^BP (x,dy),

which shows that, for anyx∈X₀∩X₁,^BP is a regular conditional probability with respect toBand^BP (x,·).

(ii)⇒(i) Assume there exists a measurable subsetX₂ofXsuch thatμ(X₂)=1 and for anyx∈X₂,

∀A∈X ^BP (y, A)=E^B_{P (x,·)}[1A|B](y) for^BP (x,·)-almost everyy∈X.

Then forx∈X0∩X2, we have, for anyB∈B,

BP (x, B)=^BP (y, B)=E^BP (x,·)[1B|B](y)=1B(y) for^BP (x,·)-almost everyy∈X.

Hence the assertion (i).

To prove the last assertion of the proposition we need the following lemma.

Lemma 4.1. LetBandCbe two sub-σ-algebras ofXsuch thatC⊂B.Then forμ-almost everyx∈X,we have the probability equalities

CP^BP (x,·)=^BP^CP (x,·)=^CP (x,·).

Proof. For anyA∈X, we have the classicalμ-almost everywhere equalities Eμ Eμ[1A|B]|C

=Eμ Eμ[1A|C]|B

=Eμ[1A|C].

Moreover, iff andgare non-negative (or bounded) measurable functions, we know that:f =g μ-a.e.⇒Eμ[f|B] = Eμ[g|B]μ-a.e. It follows that, for anyA∈X,

CP^BP (x, A)=^BP^CP (x, A)=^CP (x, A) forμ-almost everyx∈X.

Then the result follows from Proposition 2.2.

Assume (ii), forμ-almost everyz∈X, we have: for any(A, B)∈X×B,

X

1A(y)1B(y)^BP (z,dy)=

X

BP (y, A)1B(y)^BP (z,dy).

Asμ(dz)=

X

CP (x,dz)μ(dx), forμ-almost everyx∈Xand^CP (x,·)-almost everyz∈X, for any(A, B)∈X×B,

X

1A(y)1B(y)^BP (z,dy)=

X

BP (y, A)1B(y)^BP (z,dy).

Integration by^CP (x,dz)gives us, forμ-almost everyx∈X,

∀(A, B)∈X×B

X

1A(y)1B(y)^CP^BP (x,dy)=

X

BP (y, A)1B(y)^CP^BP (x,dy).

Then the result follows from Lemma 4.1.

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4.3. Proof of Theorem3.4 Case of twoσ-algebras We need the following result.

Theorem 4.2. Let(Ω,F,P)be a probability space.LetF1[resp.F2]be a sub-σ-algebra ofF;we callP1[resp.P2] the operator of conditional expectation relative toF1[resp.F2]on the spaceL¹(Ω,F,P).

Then,forf ∈L¹(Ω,F,P),the sequences of functions

1 n

n−1

k=0

(P1P2)^kf

n≥1

and

1 n

n−1

k=0

(P2P1)^kf

n≥1

convergeP-almost everywhere and in normL¹(P)towardsE_P[f|F₁∩F₂].

Proof. It’s a consequence of the classical ergodic theorem of E. Hopf (see [5], Proposition V-6-3). To identify the limit we note that:P2P1is the dual operator ofP1P2and, as the operatorsP1andP2are idempotent, the common

limit isP1- andP2-invariant (see also [1]).

LetB1andB2be two sub-σ-algebras ofXdecomposingμin an ergodic way which satisfyB1∩B2=B1∩B2. We setC=B1∩B2.

The above theorem tells us that, for any f ∈L¹(X,X, μ)the sequence of functions(¹_nn−1

k=0(^B¹P^B²P )^kf )_n≥1

convergesμ-a.e. and inL¹(μ)-norm towards^CPf =^CPf. Asμ(·)=

X

CP (x,·)μ(dx), it follows that, for anyf ∈L¹(X,X, μ)and forμ-almost everyx∈X, the sequence of functions(_n¹_n₋₁

k=0(^B¹P^B²P )^kf )n≥1converges^CP (x,·)-a.e. towards^CPf.

We callX0 a measurable subset ofX, such that μ(X0)=1 and for any x∈X0, fori∈1,2, ^BⁱP is a regular conditional probability with respect toBiand^CP (x,·)(Proposition 3.3).

The same theorem tells us that, for any x∈X0 and for any f ∈L¹(X,X,^CP (x,·)), the sequence of functions (¹_n_n₋₁

k=0(^B¹P^B²P )^kf )_n_≥₁converge^CP (x,·)-a.e. and inL¹(^CP (x,·))-norm towardsE^C_{P (x,}_·₎[f|B₁∩B₂]where, for i=1 or 2,B_i is the^CP (x,·)-completedσ-algebra ofB_i.

LetX be a countable subalgebra ofXgeneratingX. From above and Lemma 3.2, it follows that, forμ-almost any x∈X, for anyA∈X,

for^CP (x,·)-almost everyy∈X, ^CP (x, A)=^CP (y, A)=E^C_{P (x,}_·₎[1A|B₁∩B₂](y).

We deduce that, forμ-almost everyx∈X,

∀(A, C)∈X×(B1∩B2)

X

1A(y)1C(y)^CP (x,dy)=

X

CP (y, A)1C(y)^CP (x,dy).

These equalities extend to the couples(A, C)∈X×(B1∩B2)(Proposition 2.2).

SinceC=B1∩B2⊂B1∩B2, the above equalities show that, forμ-almost everyx∈X,^CP is a regular conditional probability with respect toC and^CP (x,·). From the Proposition 3.3, the σ-algebraCdecomposesμin an ergodic way.

Case of a sequence ofσ-algebras

Let(B_n)_n_≥₁be a sequence of sub-σ-algebras ofXwhich decomposeμin an ergodic way and satisfy the hypothesis of Theorem 3.4.

For anyn≥2, we have

1≤i≤n

Bi⊂

1≤i≤n−1

Bi∩Bn⊂

1≤i≤n

Bi.

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From our hypothesis, it follows that

1≤i≤n

B_i=

1≤i≤n

B_i

and consequently

1≤i≤n

B_i=

1≤i≤n−1

B_i∩B_n=

1≤i≤n

B_i.

We set

∀n≥1 Cn= n

k=1

Bk and C=

k≥1

Bk.

From the case treated previously, we prove by induction that, for anyn≥1, theσ-algebraC_n decomposesμin an ergodic way. From Proposition 3.3, forμ-almost everyx∈X,

∀(A, C)∈X×C_n

X

X C_n

P (y, A)1C(y)^CP (x,dy).

The decreasing martingale theorem implies that, for anyA∈Xand forμ-almost everyx∈X,Eμ[1A|Cn](x) −→

n→+∞

Eμ[1A|C](x). Consequently, for anyA∈Xand forμ-almost everyx∈X,^CⁿP (x, A) −→

n→+∞

CP (x, A).

Asμ(·)=

XCP (x,·)μ(dx), it follows that: for anyA∈Xand forμ-almost everyx∈X, for^CP (x,·)-almost everyy∈X, ^CⁿP (y, A) −→

n→+∞

CP (y, A).

While limiting itself to elementsC ofC, the dominated convergence theorem implies that, for anyA∈Xand for μ-almost everyx∈X,

∀C∈C

X

CP (y, A)1C(y)^CP (x,dy).

Now from Proposition 2.2, we can permute “for anyA∈X” and “forμ-almost everyx∈X,” which shows thatC decomposesμin an ergodic way.

The preceding proof shows the following corollary which improves Shimomura’s result.

Corollary 4.3. Let(B_i)_i_∈Nbe a sequence of sub-σ-algebras ofX.If for anyn∈Ntheσ-algebra_n

i=0B_i decom- posesμin an ergodic way,then the intersection

i∈NB_i decomposesμin an ergodic way.

Case of an uncountable family ofσ-algebras We need the following lemmas.

Lemma 4.4. Let (H,·,·) be a separable Hilbert space.Let {V_i: i∈I} a noncountable family of closed vector subspaces ofH.Then there exists a countable subsetJ ofI such that

i∈IV_i=

i∈JV_i. Proof. We easily see that the orthogonal complementV^⊥ of V =

i∈IV_i inH is equal to Vect(

i∈IV_i^⊥)(the closure of the subspace generated by

i∈IV_i^⊥). We choose a dense sequence of vectors(un)n≥1of Vect(

i∈IV_i^⊥)in V^⊥. The Schmidt orthonormalization process allows us to extract a maximal orthonormal system; that is, an Hilbert basis(e_n)_n_≥1deV^⊥. For anyp≥1,e_pis a finite linear combination of vectors from{u_n: n≥1}; eachu_n is itself a finite linear combination of vectors of

i∈IV_i^⊥. Therefore, there exists a countable subsetJ ofI such that,∀p≥ 1, ep∈Vect(

i∈JV_i^⊥). HenceV^⊥=Vect(

i∈JV_i^⊥)andV =

i∈JVi.

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Lemma 4.5. LetBbe a sub-σ-algebra ofXwhich decomposesμin an ergodic way.LetCbe a sub-σ-algebra of Bsuch that,for any boundedB-measurable functionf,there exists a boundedC-measurable functiongsatisfying f =g μ-a.e.

ThenCdecomposesμin an ergodic way.

Proof. From Proposition 3.3, forμ-almost everyx∈X,^BP is a regular conditional probability with respect toBand

CP (x,·), that is,

∀(A, B)∈X×B

X

1A(y)1B(y)^CP (x,dy)=

X

BP (y, A)1B(y)^CP (x,dy).

In these equalities, we have to replace^BP (y, A)by^CP (y, A). From Proposition 2.2, we can restrict our equalities toA∈X a separable sub-algebra ofXgeneratingX.

Letf be a boundedB-measurable function. There exists a bounded C-measurable functiong such thatf =g μ-a.e. Then we have,μ-a.e.,

Eμ[f|B] =f=g=Eμ[g|C] =Eμ[f|C]. It follows that, forμ-almost everyy∈X,

∀A∈X ^BP (y, A)=^CP (y, A).

Now fromμ(dy)=

X

CP (x,dy)μ(dx)we deduce that, forμ-almost everyx∈X, for^CP (x,·)-almost everyy∈X,

∀A∈X ^BP (y, A)=^CP (y, A) and consequently, forμ-almost everyx∈X,

∀(A, B)∈X×B

X

1A(y)1B(y)^CP (x,dy)=

X

CP (y, A)1B(y)^CP (x,dy).

Hence the result.

Lemma 4.6. Let {B_n: n∈N^∗} be a sequence of sub-σ-algebras of X satisfying, for any n≥2,

1≤k≤nB_k =

1≤k≤nB_k. Then

n≥1B_k=

n≥1B_k.

Proof. Letf ∈L¹(X,X, μ). From the decreasing martingale theorem, we have,μ-almost everywhere, Eμ

f

k≥1

B_k

= lim

n→+∞Eμ

f

1≤k≤n

B_k

= lim

n→+∞Eμ

f

1≤k≤n

B_k

= lim

n→+∞Eμ

f

1≤k≤n

Bk

=Eμ

f

k≥1

Bk

. (2)

Hence the result.

Consider the separable Hilbert space L²(X,X, μ). It is well known that, for each sub-σ-algebra B of X, the spaceL²(X,B, μ)is identified to a closed subspace ofL²(X,X, μ)and the conditional expectation relative toBis identified to the orthogonal projection onto this closed subspace.

Let{B_i: i∈I}be an uncountable family ofμ-complete sub-σ-algebras which decomposeμin an ergodic way and satisfy the hypothesis of Theorem 3.4. From the Lemmas 4.4 and 4.6, there exists a countable subsetJ ofI such that

L²

X,

i∈I

B_i, μ

⊂

i∈I

L²(X,B_i, μ)=

i∈J

L²(X,B_i, μ)=L²

X,

i∈J

B_i, μ

.

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It follows that for anyf ∈L²(X,

i∈IB_i, μ)there exists a functiong∈L²(X,

i∈JB_i, μ) such thatf =g, μ-a.e. According to the case treated previously, we know that the sub-σ-algebra

i∈JB_idecomposesμin an ergodic way. Then the result follows from Lemma 4.5.

5. Examples and applications

1. Let(X,X, μ)be a probability space with a separableσ-algebraXcontaining a μ-approximating compact class.

Letτ be an invertible bi-measurable transformation of(X,X)such thatμis quasi-invariant for the action ofτ. We consider the sub-σ-algebraJ=J_τ of Xdefined by J= {B∈X: τ⁻¹(B)=B}. Then the following result is well known.

Proposition 5.1. Theσ-algebraJdecomposesμin an ergodic way.

Proof. An idea of the proof is the following. We consider the contractionT ofL¹(X,X, μ)defined by Tf (x)=f◦τ⁻¹(x)d(τ (μ))

d(μ) (x).

Replacingτ byτ⁻¹we obtain the inverse operatorT⁻¹.

From the Chacon–Ornstein ergodic theorem, one proves [2] that, with obvious notations: for anyf∈L¹(X,X, μ) and forμ-almost everyx∈X,

n

k=−n

T^kf (x) n

k=−n

T^k1(x) −→

n→+∞

JPf (x).

One sees easily that there exists a measurable subsetX0ofXsuch thatμ(X0)=1 and for anyx∈X0the probability^JP (x,·)isτ-quasi-invariant with^d(τ_d_J^J_{P (x,·)}^{P (x,}^·⁾⁾ =^{d(τ μ)}_dμ .

Then the same ergodic theorem tells us that, for anyx∈X0, for anyf ∈L¹(X,X,^JP (x,·))and for^JP (x,·)-almost everyy∈X,

n

k=−n

T^kf (y) n

k=−n

T^k1(y) −→

n→+∞E^J_{P (x,}_·₎[f|J](x).

As in the first case of Theorem 3.4, we prove that forμ-almost everyx∈X,^JP is a regular conditional probability with respect toJand^JP (x,·). The result follows from Proposition 3.3.

In [3], Greshchonig and Schmidt consider the case of a Borel action of a locally compact second countable group Gon a standard probability space(X,X, μ); that is, a group homomorphismg→τ_gfromGinto the group Aut(X) of Borel automorphisms ofXsuch that the map (g, x)→τ_gx fromG×XtoX is Borel andμ is quasi-invariant under eachτg,g∈G. They prove that theσ-algebra

g∈GJτ_g decomposesμin an ergodic way.

The Theorem 3.4 makes it possible to find and improve this result.

Corollary 5.2. Let{τ_i: i∈I}be a non-empty family of Borel automorphisms ofX.Then the sub-σ-algebra

i∈IJ_τ_i ofXdecomposesμin an ergodic way.

Proof. Taking into account the Proposition 5.1, it is enough to show that, for any finite subsetJ ofI,

i∈JJ_τ_i=

i∈JJτ_i. Letf be a

i∈JJτ_i-measurable function. We setX0=

i∈J{f◦τi=f}; we haveμ(X0)=1.

We callGthe algebraic subgroup of Aut(X)generated by the Borel automorphisms{τ_i: i∈J};Gis a countable subset of Aut(X). The subsetX₁=

s∈GsX₀ofX₀belongs to

i∈JJ_τ_iandμ(X₁)=1. Then the functiong=f1X1

is

i∈JJτ_i-measurable andf=gμ-a.e. Which shows thatf is

i∈JJτ_i-measurable.

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2. Let(X,X, μ, τ )be a dynamical system with a polish space and a not necessarily invertible transformation. We denote by (Y,F, λ, η) the natural extension of our dynamical and byπ the natural projection of Y ontoX. With obvious notations, one sees easily thatf isJ_η∩π⁻¹(X)-measurable (resp.J_η∩π⁻¹(X)-measurable) if and only if there existsg∈J_τ such thatf =g◦π (resp.f =g◦π λ-a.e.). It follows that

J_η∩π⁻¹(X)=J_η∩π⁻¹(X).

We know that theσ-algebraJ_ηdecomposesλin an ergodic way. Theσ-algebraπ⁻¹(X)is separable. Therefore theσ-algebraC=J_η∩π⁻¹(X)decomposesλin an ergodic way.

LetP be a regular conditional probability with respect toJ_τ andμ. LetQbe a regular conditional probability with respect toCandλ. For anyA∈XandC∈J_τ we have:

X

P (x, A)1C(x)μ(dx)=

X

1A(x)1C(x)μ(dx) and therefore

Y

P

π(y), A 1C

π(y)

λ(dy)=

Y

1A

π(y) 1C

π(y) λ(dy)

=

Y

Eλ[1A◦π|C](y)1C

π(y) λ(dy)

=

Y

Q

y, π⁻¹(A) 1C

π(y)

λ(dy). (3)

Which proves, via the Proposition 2.2, that forλ-almost everyy∈Y, P

π(y),·

=Q

y, π⁻¹(·) and theσ-algebraJ_τ decomposesμin an ergodic way.

3. Let P be a transition probability on a measurable space (X,X) with a separable σ-algebra Xcontaining a μ-approximating compact class.

We denote byΠ the set ofP-invariant probability measures on(X,X):

π∈Π ⇔

X

f (x)π P (dx)=

X

Pf (x)π(dx)=

X

f (x)π(dx)

for any non-negative or bounded measurable functionf onX. We assume thatΠ=∅.

For anyπ∈Π, we denote byB_π the sub-σ-algebra ofXdefined by:

Bπ= {A∈X: P1A=1Aπ-a.e.} and we setB=

π∈ΠB_π.

Letπ∈Π. Letf be a boundedB_π-measurable function onX. The functiong, defined by g(x)=lim inf1

n

n−1

k=0

P^kf (x),

satisfiesg=f π-a.e. andP g≤g. From the latter inequality, it follows that, for anyσ ∈Π,P g=g σ-a.e. andgis B-measurable. We deduce thatB_π=Bπ-a.e.

From the Hopf theorem ([3], Proposition V-6-3), for anyf ∈L¹(X,X, π ), the sequences of functions

1 n

n−1

k=0

d((f π )P^k) dπ

n≥1

and

1 n

n−1

k=0

P^kf

n≥1

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convergeπ-almost everywhere and in norm L¹(π )towards ^B^πPf =^BPf. As in Example 1, one deduces thatB decomposesπin an ergodic way.

Acknowledgments

I wish to thank B. Bekka and S. Gouëzel for helpful discussions and the referees for their careful reading and valuable comments.

References

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