数理解析研究所講究録 1215 巻 2001 年 33-52

33

WHAT DIFFERENTIATES STATIONARY STOCHASTIC
PROCESSES FROM ERGODIC ONES: ASURVEYMICHEL VALADIER
ABSTRACT. Stationarity of astochastic process seems connected to the idea of constancy. But ergodicity is needed for the property that almost surely the observation
to 0makes possible the identification of the law of theof atrajectory from time
whole process, including the future. When the stationary process is aMarkov chain
with afinite number of states it is well known that the set of states divides in ergodic
classes. Decomposition of more general stationary processes in ergodic classes goesback to von Neumann. This seems ahidden result rarely developed in t\.{e}xt books.
After some preliminaries we will expose Choquet way and the Kryloff-Bogoliouboff
way which was made abit more precise by Oxtoby and greatly generalized by Dynkin.
$-\infty$

1. Introduction
Very often in contemporary papers the authors assume that astochastic process isergodic because under the weaker hypothesis of stationarity their proofs no longer

hold. One aim of this paper is to show that this difficulty may be immaterial.
Aspecially interesting problem is prediction. Stationarity (of astochastic process–for aprecise definition see Section 2) seems connected to the idea of constancy.

But ergodicity is needed for the property that (almost surely of course) the observation of atrajectory from time -co to 0makes possible the identification of thelaw of the whole process, including the future: this is made precise in Theorem 2
and the consequence after.
When the stationary process is aMarkov chain with afinite number of states
the classical theory, already in [Do, Chapter$\mathrm{V}$], shows that the set of states divides, 60.
1991 Mathematics Subject Classification. 60-02, 60
Key words and phrases, stationarity, ergodicity, Birkhoff’s theorem, prediction.
$\mathrm{G}10$$\mathrm{G}25$

34
M. VALADIER

in ergodic classes. And when one observes atrajectory, this trajectory lives in one

ergodic class and it amounts (if only one observation is done which is the only

possibility if the time is the one of our life) to the same thing as if the process

was ergodic. Decomposition of more general stationary processes in ergodic classes
goes back to von Neumann [N1]. This seems ahidden result rarely developed intext books: Denker, Grillenberger and Sigmund’s book [DGS] is an exception but
it does not emphasize the consequences. All amounts to the following: there isaprobability law which Icall it contingency law,

firstly an ergodic law
according to

$Q$

$Q$

$\lambda$, on the set of ergodic laws;

is chosen according to Aand then the trajectory is chosen.

After some preliminaries we will give the main result. It was proved by different
methods. The elegant Choquet way will be summarized quickly. The KryloffBogoliouboff way which was made abit more precise by Oxtoby and greatly gener-

alized by Dynkin will be more detailed. For some other works see Maharam [Mah],
Varadarajan [Var] and for ageneralization to capacities Talagrand [T].

Ihave discovered the existence of Chersi’s paper [Che] in May 2000. Idon’tknow more than the title and the very short Math. Reviews analysis. Surely it isalso on the subject.
The author thanks Gerard MICHAILLE and Charles CASTAING for their helpfulcomments.

2. Stationary and ergodic processes

Let$(K, \mathcal{K})$be aBorel standard measurable space, that is ameasurable space is0-

morphic to aBorel subset of aPolish topological space (for the Choquet point of35
STATIONARY PROCESSES
view, $K$ will be compact metrizable). Astochastic process with discrete time tak-ing its values in $K$ is abilateral sequence$\mathrm{r}.\mathrm{v}.)$set

defined on aprobability space$\Omega=K^{\mathrm{Z}}$is more fundamental$\mathcal{F}=B(\Omega)$$\mathcal{F}$of random variables (in shortwhich take their values in $K$ . The$(_{-}^{-}-, S, \Pi)$$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{n}---$

metrizable when $K$ is compact. Let$(X_{n})_{n\in \mathrm{Z}}$. It is still Borel standard and is compact$=\mathcal{K}^{\otimes \mathrm{Z}}$; when $K$ is compact and $\mathcal{K}=B(K)$ ,, that is the product of the Borel tribes coincides with the Borel tribe of

the product topology. The law of the process, always denoted by $P$ in this paper, isthe probability measure on

process$(\overline{X}_{n})_{n\in \mathrm{Z}}$$(\Omega, \mathcal{F})$is defined on$\Omega$

image of

Apointdefined by$(X_{n})_{n\in \mathbb{Z}}$$\omega$$(_{-}^{-}-, S, \Pi)$$=(x_{n})_{n\in \mathrm{Z}}\in\Omega$$\tilde{X}_{n}$is the -th coordinate$n$and the canonical process is simply$(X_{n})_{n\in \mathrm{Z}}$$(X_{n})_{n\in \mathbb{Z}}$$\Omega$in itself

is the Bernoulli shift. It is an homeomorphism$T$is denoted by $T_{\#}(P)$ .

is stationary if its law is invariant i.e. for any, $P(T^{-1}A)=P(A)$ (that is $T_{\#}(P)=P$ ). We write

Definitions. Let. The “canonical”

is atrajectory. The bijective map $T$ of$T((x_{n})_{n\in \mathrm{Z}})=(x_{n+1})_{n\in \mathrm{Z}}$Definition. The process$A\in \mathcal{F}$$\xi\mapsto(X_{n}(\xi))_{n\in}\mathrm{z}$.

when $K$ is compact. The image of $P$ by$A\in \mathrm{B}(\mathrm{K})$byby the property thatfunction. In the sequel we do not use

denoted by$\Pi$$P\in \mathcal{L}_{\mathrm{s}\mathrm{t}}$.

be astationary process. The invariant events are thesatisfying $T^{-1}A=A$ (or $TA=A$ ). The set they constitute is atribe

denoted by I. The process of law $P$ is $ergodic^{1}$ if Iis coarse (up to P-negligiblesets), that is if $A\in \mathrm{I}\Rightarrow P(A)=0$ or 1. One also says that $P$ is an ergodic law.

We write$P\in \mathcal{L}_{\mathrm{e}\mathrm{r}\mathrm{g}}$.

In ergodic theorems,
1$T^{j}$

denotes, when$j\in \mathbb{Z}_{+}^{*}=\mathrm{N}^{*}$, the -th power of $T$ :Doob [Do, p.457], and several authors, say “metrically transitive”.

$j$

36
M. VALADIER$T^{j}=T\mathrm{o}T\mathrm{o}\cdots \mathrm{o}T$The notation$\delta_{x}$,$T^{0}=\mathrm{i}\mathrm{d}\mathrm{n}$and, when$j\in \mathbb{Z}_{-}^{*}$

denotes the Dirac measure at,$T^{j}$$x$denotes$(T^{-1})^{|j|}$.

.

3. Identification of the law of an ergodic process from the observationof its past
Proposition 1is elementary. It will be applied, when $K$ is compact, toas well as to$\mathrm{A}=K^{d}$$\mathrm{A}=\Omega=K^{\mathrm{Z}}$. For the Borel standard case see Proposition 5.

Proposition 1. Let Abe compact metrizable and $D$ a dense $subset^{2}$ ofally D ill be countable). Let$(P_{k})_{k\in \mathrm{N}}$$\mathrm{C}(\Lambda)$

be a sequence of probability measures on

It weakly converges (i.e. for the weak topology relative to the duality with$\forall f\in D$, the sequence$( \int_{\Lambda}fdP_{k})_{k\in \mathrm{N}}$$( \int_{\Lambda}fdP_{k})_{k\in \mathrm{N}}$converges in R. The space$\mathcal{M}_{+}^{1}(\Lambda)$

sures on Ais weakly compact metrizable. The sequence
point. Indeed if$Q_{1}$

and$Q_{2}$$\int_{\Lambda}fdQ:=\lim_{k}\int_{\Lambda}fdP_{k}$$\mathrm{C}(\Lambda)$$\Lambda$

.

) iff

converges in R.

Proof The “only if” part is obvious. For the converse assume thatsequence(usu-$\forall f\in D$of all probability mea-$(P_{k})_{k\in \mathrm{N}}$has aunique limit

are two limit points, one has, for $i=1$ and 2,

hence $Q_{1}=Q_{2}$ .

, the$\forall f\in D$,$\square$

Theorem 2. Assume that $K$ is compact and that the process $(Xn)nez$ with values
in $K$ is ergodic.
$(X_{-d+1}, \ldots, X_{0})$Then almost surely, for allis the weak limit of$d\in \mathrm{N}^{*}$$k^{-1} \sum^{k-1}j.=0\delta(x_{-(j+d-1)},\ldots,x_{-j})$Consequence. Hence $P$-almost surely, knowing$P_{(X_{-d+1},\ldots,X_{0})}$$p\leq q)$

hence of

$P(X_{\rho},X_{\mathrm{p}+1},\ldots,X_{q-1},X_{q})$since stationarity implies, the law $P(x_{-d+1},\ldots,x_{0})$$(x_{n})_{n\leq 0}$$C(\Lambda)$

as $karrow+\infty$ .

implies the knowledge of(this writing supposes$P(X_{p},\ldots,X_{q})=P_{(X_{\mathrm{p}-q},\ldots,X_{0})}$2 It is sufficient that the linear subspace ofof$(p, q)\in \mathbb{Z}^{2}$and. Recall now thatspanned by $D$ is dense

$P$

37
STATIONARY PROCESSES

is the projective limit of the measures3$P_{(X_{p},\ldots,X_{q})}$. So, mathematically, $P$ can be

identified; from anumerical point of view, this is another story: see all the concepts

defined and studied in Statistical Theory.

Proof. Let
let

$f$

$d\in \mathrm{N}^{*}$$D_{d}$

be acountable dense subset of

denote the function on$f_{0}(x_{-d+1}, \ldots, x_{0})$set$\mathrm{b}\mathrm{l}\mathrm{e}$and$N_{f_{0}}$$\Omega$associated to. By Birkhoff’s theorem if$f\mathrm{o}$$\mathrm{C}(K^{d})$. For any

which is defined by:$(x_{n})_{n\in}\mathrm{z}$$f_{0}\in D_{d}$$f((x_{n})_{n\in \mathrm{Z}})=$

does not belong to aP-negligi-:$\int_{\Omega}f_{0}d[k^{-1}$I$\delta_{(x_{-(j+d-1)}},\ldots$, $x_{-j})]=k^{-1} \sum_{j=0}^{k-1}f_{0}(x_{-(j+d-1)}, \ldots, x_{-j})$$=k^{-1} \sum_{j=0}^{k-1}f[T^{-j}((x_{n})_{n})]$$\underline{(karrow+\infty)}\int_{\Omega}fdP$$= \int_{K^{d}}f_{0}dP_{(X_{-d+1\prime}\ldots,X_{0})}$By Proposition 1this proves the convergence of$P_{(X_{-d+1},\ldots,X_{0})}$if$(x_{n})_{n\in \mathbb{Z}} \not\in\bigcup_{f\mathrm{o}\in D_{d}}N_{f\mathrm{o}}$$\bigcup_{d\in \mathrm{N}^{\mathrm{r}}}[\bigcup_{f\mathrm{o}\in D_{d}}N_{f\mathrm{o}}]$

.

$.$

.

$k^{-1} \sum_{j=0}^{k-1}\delta_{(x_{-(\mathrm{j}+d-1)},\ldots,x_{-j})}$-So the statement holds for$(x_{n})_{n\in \mathrm{Z}}$tonot in$\square$Comment. The meaning of Theorem 2is that almost surely the mere observationof the past$(\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}-\infty)$of an ergodic process allows to identify the law of the wholeprocess (including the future). From atheoretical point of view this is aperfectsituation for prediction. Indeed whenthat is$\xi=(x_{n})_{n\leq 0}=\omega_{|\mathrm{Z}_{-}}$

and ($\Omega$

is written$=(x_{n})_{n>0}=\omega_{|\mathrm{N}}.$(see next Section) of $P$ unique up to equality
3$\Omega=K^{\mathrm{Z}_{-}}\cross K^{\mathrm{N}}$$\mathrm{a}.\mathrm{e}$. and$\omega$$=(\xi, \zeta)$, there is adisintegration. which is afamily$(L_{\xi})_{\epsilon\epsilon\kappa^{\mathrm{z}_{-}}}$ofMore simply when theare known, the values of $P$ on the algebra of cylindrical.
sets are known, and this algebra generates$P(\lambda_{p}’, .,X_{q})$$B(\Omega)$

38
M. VALADIER

probability laws on

the conditional law$K^{\mathrm{N}}$. Then knowingon$L_{\xi}$$K^{\mathrm{N}}$$\xi=\omega_{|\mathrm{Z}_{-}}=(x_{n})_{n\leq 0}$, the future obeys to.

4. Basic ideas of disintegration

When asub-tribe$\mathcal{G}$potheses concerning

ability measuresof$\Omega$$\mathcal{F}$is given, there exists under very general topological hy-, adisintegration with respect to , that is afamily of prob$\mathcal{G}$$(Q^{\cdot})_{\omega\in\Omega}$$\forall B\in \mathcal{F}$,on$(\Omega, \mathcal{F})$$\forall A\in \mathcal{G}$,which is -measurable in$\mathcal{G}$and which satisfies$\omega$$P(A \cap B)=\int_{A}Q^{\omega}(B)dP(\omega)$.

Let us consider, as it always should be, the conditional expectation
class of random variables up to equality
through$\mathcal{F}$$\mathrm{a}.\mathrm{s}$. The functions$\omega$$\mathrm{E}^{\mathcal{G}}(1B)$$\mapsto Q^{\omega}(B)(B$) constitute a“consistent”4 family of versions of the$\mathrm{E}^{\mathcal{G}}(1B)$

as arunning.

This has along story in probability theory: von Neumann [N1], Kolmogorov[Ko], Jirina [Ji], Hoffmann-Jorgensen [HJ], Valadier [V1-2] for some$\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{i}1\mathrm{s}^{5}$. (Fortextbooks see Bauer [Ba], Dudley [Du].) But disintegration is unduly considered asahard concept reserved to experts and, in my opinion, too rarely used.
Classically for any real integrable$\mathrm{r}.\mathrm{v}$.

(see for example Dudley [Du, 10.2.5$Y$

p.272], Doob [Do, Th.9.1p.27], Kolmogorov [Ko, ch.V (12) and (14)] :$)$.

$\int_{\Omega}Y(\omega’)dQ^{\omega}(\omega’)\mathrm{a}.=\mathrm{s}$$(\mathrm{E}^{\mathcal{G}}Y)(\omega)$.

An important particular case is the following. Suppose$\omega$$=$$(\xi, \langle)$and$\mathcal{G}$is generated by the projection on

4 The problem if one chose anyhow versions of$\mathrm{E}^{Q}(1B)$$\Omega_{1}$$\Omega$is a(possibly$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}^{6}\Omega_{1}\cross\Omega_{2}$

$\xi$

,

is the past, (iswould lie in the -additivity with respect$\sigma$to . Aclassical expression for disintegration is regular conditional probabilities.
5 At the time when in France only Bourbaki and Jirina were quoted, Iwrote [V1] where Igave$B$

aresult of Hoffmann-Jorgensen [HJ] in the framework of aproduct and where Icompared several
statements of this time. In [V2, p. 13] Ihad the idea, being not aware of [Hll], of introducing thequotient tribe.
6 To be more precise
is a“good”
is separated and countably generated andtopological space, that is Polish or Suslin...
$(\Omega_{1}, F_{1})$$\Omega_{2}$39
STATIONARY PROCESSES
the future). Then

$Q^{\omega}$

Dirac mass at 4anddepends only on 4and has the form

$L_{\xi}$

is the conditional law of$\langle$

given

$\xi$

$\delta_{\xi}\otimes L_{\xi}$where$\delta_{\xi}$is the.

5. Decomposition of astationary process. The contingency law

In the following$P$

always denote aprobability measure on$\Omega=K^{\mathrm{Z}}$and we will

say equivalently that $P$ is invariant or stationary. This refers to the stationarity of
the “canonical” process defined in Section 2. And $P$ is said ergodic if the process
is ergodic. Although the decomposition theorem admits several non trivial proofs

and some variants in its formulation, it roughly says at least the following:
Theorem 3. Any stationary law $P$ on$\Omega$

is a mixing of ergodic laws.

Comments. 1) All amounts to the following: there is aprobability law which Icall
contingency law,$\lambda$, on the set of ergodic laws; firstly an ergodic law

according to Aand then the trajectory is chosen according to

$Q$

$Q$

is chosen. So, if only one

observation is done, one observes atrajectory of an ergodic process. And in myopinion, the prediction of stationary$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{s}^{7}$

is not aproblem different from theprediction of ergodic ones.
2) For example imagine the set of meteorological phenomena appearing during

one year is the value of astationary process with time in$\mathbb{Z}$, and imagine that thisprocess has been always observed. Then it could be treated as an ergodic process:two moons or another rotational velocity of the planet Earth could have occurred

if the world has been created differently. This is contingency.
3) Any probability is amixing of Dirac measures: if$P\in \mathcal{M}_{+}^{1}(\mathbb{R})$it is the mixing

7 On the subject of prediction of stationary processes there is avery ambitious book:

Fursten

40
M. VALADIERof the measures$\delta_{r}$according to the image$\overline{P}$

of

on

$P$

by

$\mathcal{M}_{+}^{1}(\mathbb{R})$$r\mapsto\delta_{r}$. This has

not any interest. In the interpretation of Theorem 3the roles of time, past andfuture are essential.
Theorem 3was originally proved by von Neumann in 1932 [N1]. It received
several interesting proofs: the one of Choquet [Choi, 1956/1957], the method of

Kryloff-Bogoliouboff [KB, 1937]; see also Mahaxam [Mah, 1950], Farrell [Fa, 1962],
and specially Dynkin [Dy, 1978].

5.1 Choquet’s way.

Assume $K$ is compact metrizable. Letsures on$(\Omega, B(\Omega))$$\mathcal{M}_{+}^{1}(\Omega)$

denote the set of probability mea-

endowed with the weak topology.

Theorem 4. Assume $K$ is compact metrizable. 1) The set

abilities on$\Omega$

is a non empty convex compact subsetof extreme points of$\mathcal{L}_{\mathrm{s}\mathrm{t}}$of$\mathcal{M}_{+}^{1}(\Omega)$

coincide with the set of ergodic laws

There exists a probability measure Aon$\partial \mathcal{L}_{\mathrm{s}\mathrm{t}}=\mathcal{L}_{\mathrm{e}\mathrm{r}\mathrm{g}}$$P= \int_{\mathcal{L}}.Qd\lambda(Q)r_{C}$$\mathcal{L}_{\mathrm{s}\mathrm{t}}$of invar iant prob. 2) The set$\mathcal{L}_{\mathrm{e}\mathrm{r}\mathrm{g}}$. 3) Let$\partial \mathcal{L}_{\mathrm{s}\mathrm{t}}$$P\in \mathcal{L}_{\mathrm{s}\mathrm{t}}$.

such that.

(1)(In (1) the right-hand side is aweak integral of measures whose meaning is as well$\forall\varphi\in \mathrm{C}(\Omega)$,$\int_{\Omega}\varphi dP=\int_{\mathcal{L}_{\mathrm{e}\mathrm{r}\mathrm{g}}}[\int_{\Omega}\varphi dQ]d\lambda(Q)$as$\forall B\in B(\Omega)$,$P(B)= \int_{\mathcal{L}_{\mathrm{e}\mathrm{r}\mathrm{g}}}Q(B)d\lambda(Q).)$

41
STATIONARY PROCESSES

Some ideas of the proof. The first assertion has an easy proof. The second is wellknown of specialists and arather old result: see Blum-Hanson [BH, 1960] and ChOquet knew it before; Irecommend the proof of Denker-Grillenberger-Sigmund [DGS,(5.6) p.24]. Then the conclusion follows ffom the Choquet integral representation

theorem (besides the quoted works of Choquet one can see [Bo, IV.7.2 Th.l p.219]
and Phelps [Ph] .
$)$Remarks. One year before Choquet, Hewitt-Savage [HS] used the same argumentwith the set of laws on$\mathbb{R}^{I}$

invariant by permutation of coordinates whose extreme

points are the laws of families of i.i.d. random variables (they continued afamouswork of de Finetti [Fi] . But in [HS] the set of extreme points is closed which$)$makes the integral representation elementary while here
stationary Markov chains with matrices($\frac{1}{n}\frac{1}{n}$$1 \frac{1}{n}\frac{1}{-n}$$\mathcal{L}_{\mathrm{e}\mathrm{r}\mathrm{g}}$is not closed (think of), for$n\in \mathrm{N}^{*}$, $narrow+\infty$ ).

5.2 Kryloff-Bogoliouboff’s way (generalized by Dynkin).
Let us summarize the main result of Kryloff-Bogoliouboff as it was generalized by
Dynkin [Dy].

Proposition 5. Let$\Omega$be Borel standard.

There exists a countable set $W$ ofbounded measurable functions satisfying:$\mathrm{B}\mathrm{S}1$. Let$(P_{k})_{k\in \mathrm{N}}$the sequence$( \int_{\Omega}fdP_{k})_{k\in \mathrm{N}}$That$\forall f\in W$$\mathrm{B}\mathrm{S}2$. If$\mathcal{H}$be a sequence of probability measures on,$\Omega$such That $\forall f\in W$ ,converges in R. Then there exists a probability$\int_{\Omega}fdP_{k}arrow\int_{\Omega}fdP$

$P$

such

.

is a linear space of real functions on$\Omega$

containing $W$ and stable withrespect to the “bounded pointwise $convergence^{8}$ ”, then $H$ contains all bounded mea8 That is: if $f_{n}\in$ }{, if the sequence $(f_{n})_{n}$ is uniformly bounded and converges pointwisely to42
M. VALADIER

surable functions on$\Omega$.

Thanks to BS2 there is auniquesatisfying$P$$\mathrm{B}\mathrm{S}1$.

Remark. Ashort time after Dynkin’s paper [Dy, 1978], Ramakrishnan and Raoproved in [RR, 1980] that the notion of $B$ -space introduced by Dynkin coincides

with the notion of Borel standard.

Proof See Dynkin [Dy] Section 4.2 page 714.
Notations. Let$\Omega$. Let$\tilde{\mathcal{F}}$$\square$denote the set of all real bounded-me surable functions on$\mathcal{F}$denote the intersection$\hat{\mathcal{F}}$$\hat{\mathcal{F}}=\cap\hat{\mathcal{F}}_{P}P\in \mathcal{L}_{*\mathrm{t}}$of the-completions of$P$

Theorem 6. (1) Let$\mathcal{F}$.

$Q_{n}^{\omega}$

two following subsets of ,$\Omega$$\Omega’:=$and, if$Q^{\omega}$$\forall f\in W$,{$\omega$denote,$\mathcal{L}_{\mathrm{s}\mathrm{t}}$$f$,and $W$ given by Proposition 5. The

defined by:$\Omega_{\mathrm{e}\mathrm{r}\mathrm{g}}$$\in\Omega:\forall f\in W$$( \int_{\Omega}fdQ_{n}^{\omega})_{n}$converges in$\mathbb{R}$}denotes the probability –whose existence follows from BS1 –satisfying$\int_{\Omega}fdQ_{n}^{\omega}arrow\int_{\Omega}fdQ^{\omega}$$\Omega_{\mathrm{e}\mathrm{r}\mathrm{g}}:=$

belong toand$\Omega’$$\frac{1}{n}\sum_{i=0}^{n-1}\delta_{T^{j}\omega}$$\hat{\mathcal{F}}$and have$\forall f\in\overline{\mathcal{F}}$, then $f\in H$ .

,

$P$

{

,

$\omega$$\in\Omega’$:$Q^{\omega}$

is an ergodic law}-measerre1for any invariant probability P. Moreover$\int_{\Omega}fdQ_{n}^{\omega}arrow\int_{\Omega}fdQ^{\omega}P- \mathrm{a}.\mathrm{s}.$.

$\forall P\in$43
STATIONARY PROCESSES(2) The family$\mathrm{I}$$(Q.)_{\omega\in\Omega_{\mathrm{e}\mathrm{r}\mathrm{g}}}$, which means: for any$\forall B\in \mathcal{F}$as well as, for any,

disintegrates any invariant probability $P$ relatively to$B\in \mathcal{F}$$\forall A\in \mathrm{I}$

,

,

.or

$(\mathrm{E}_{P}^{\mathrm{I}}(1_{B}))(\omega)=Q^{\omega}(B)P- \mathrm{a}.\mathrm{s}$(2)$P(A \cap B)=\int_{A\cap\Omega_{\mathrm{e}r_{\zeta}}}Q^{\omega}(B)dP(\omega)$$Y\in \mathcal{L}^{1}(\Omega, \mathcal{F}, P)$(or $Y\geq 0$ and F-measurable),$(\mathrm{E}_{P}^{\mathrm{I}}Y)(\omega)=P- \mathrm{a}.\mathrm{s}$.

$\int_{\Omega}YdQ^{\omega}$

Remarks. 1) When $K$ is compact metrizable as in [KB], [Ox] and [DGS], one getson aset denoted by$\Omega_{\mathrm{q}\mathrm{r}}$, the weak convergence$Q_{n}^{\omega}arrow Q^{\omega}$

for the$\sigma(\mathcal{M}^{b}(\Omega), C(\Omega))$

topology. There are less $P$-negligible sets and equalities $P$-almost everywhere thanhere (in the [Dy] framework). And here one gets only the following two weak

convergences: (i) on the set(ii)$\forall P\in \mathcal{L}_{\mathrm{s}\mathrm{t}}$$P( \{\omega\in\Omega’ :,,$\forall f\in\overline{\mathcal{F}}$$\Omega’$$Q^{\omega}$.

\int_{\Omega}fdQ_{n}^{\omega}arrow\int_{\Omega}fdQ^{\omega}\})=1$

to be bijective. When$\int_{\Omega}fdQ_{n}^{\omega}arrow\int_{\Omega}fdQ^{\omega}$$\int_{\Omega}fdQ_{n}^{\omega}arrow P- \mathrm{a}.\mathrm{s}$2) In the Dynkin framework[DGS],, $\forall f\in W$ ,$\Omega$$\Omega=K^{\otimes \mathbb{Z}}$$\int_{\Omega}fdQ^{\omega}$, that is;$\forall f\in\overline{\mathcal{F}}$,$\forall P\in \mathcal{L}_{\mathrm{s}\mathrm{t}}$,.

is not necessarily aproduct and$T$is not assumed

and $T$ is the Bernoulli shift, as in [KB], [Ox] andcan be treated as abilateral limit as written in the following formula$Q^{\omega}:= \lim_{|n|arrow+\infty}.[|n|^{-1}$$\delta_{T^{j}}(\omega)]$.

$\mathrm{s}\mathrm{g}\mathrm{n}j=\mathrm{s}\mathrm{g}\mathrm{n}n\sum_{|j|\leq|n|-1},$

Proof. All arguments are those of [Dy] with some simplifications possible thanks to
the framework and some more explanations when Ifeel them useful. Part (A) comesfrom Lemmas 4.1 and 6.1 of [Dy], (B) comes from Lemma 3.1 and Theorem 3.4 and(C) is apart of Theorem 3.1

44
M. VALADIER(A) It is easy to prove$\Omega’\in \mathcal{F}$$\forall P\in \mathcal{L}_{\mathrm{s}\mathrm{t}}$,. By Birkhoff’s theorem$\forall f\in\tilde{\mathcal{F}}$

Since $W$ is countable, (3) implies
and (3),$\forall P\in \mathcal{L}_{\mathrm{s}\mathrm{t}}$,$\forall f\in W$,.
$\int_{\Omega}fdQ_{n}^{\omega}arrow P- \mathrm{a}.\mathrm{s}$$\forall P\in \mathcal{L}_{\mathrm{s}\mathrm{t}}$$(\mathrm{E}_{P}^{\mathrm{I}}f)(\omega)$, $P(\Omega’)=1$ . By the definition of.

$P\in \mathcal{L}_{\mathrm{s}\mathrm{t}}$for;then (4) still holds if$f\in \mathcal{L}^{1}(\Omega, \mathcal{F}, P)$letter$f$

in place ofso$Tl_{P}$$f$is$H_{P}=\tilde{\mathcal{F}}$$\mathcal{F}$.

\int_{\Omega}fdQ^{\omega}\}$. As consequences, (4) holds firstly if-measurable and $[0, +\infty]$ -valued and it holds too$Y$$\mathcal{F}_{|\Omega’}$

theorem, for). Now, again by Birkhoff\int_{\Omega}fdQ_{n}^{\omega}arrow\int_{\Omega}fdQ^{\omega}\})=1$

Now we discuss the measurability properties ofisto the end(use $f=f^{+}-f^{-}$ ). This proves Part 2of the statement (with$P( \{\omega\in\Omega’ :$\omega\mapsto\int_{\Omega}fdQ^{\omega}$$\omega\mapsto Q^{\{\lrcorner}$,$H_{P}:= \{f\in\tilde{\mathcal{F}}:(\mathrm{E}_{P}^{\mathrm{I}}f)(\omega)=P- \mathrm{a}.\mathrm{s}.

$f\in\tilde{\mathcal{F}}$$Q^{\omega}$(4)$\int_{\Omega}fdQ^{\omega}$

We postpone the discussion of the measurability properties ofProperty BS2 applies to(3),$(\mathrm{E}_{P}^{\mathrm{I}}f)(\omega)=P- \mathrm{a}.\mathrm{s}$of Part (A). Now let for.

$\omega\mapsto Q^{\omega}$$f\in\tilde{\mathcal{F}}$and$P\in \mathcal{L}_{\mathrm{s}\mathrm{t}}$,.

. Firstly for any$f\in\overline{\mathcal{F}}$,-measurable because it is the limit of asequence of measurable

functions. Thanks to BS2$H$ $:=$

equals$\tilde{\mathcal{F}}${$f\in\overline{\mathcal{F}}$:. Then (4) extended tohence there exists$\omega\mapsto\int_{\Omega}fdQ^{\omega}$$\tilde{\mathcal{F}}$$\varphi B:\Omegaarrow \mathbb{R}$is$\mathcal{F}_{|\Omega’}$

implies: for any

which ismeasurable$B\in \mathcal{F}$,$\omega\mapsto Q^{\omega}$is I-measurable,-measurable and satisfies$\mathrm{J}$$\forall P\in \mathcal{L}_{\mathrm{s}\mathrm{t}j}$45
STATIONARY PROCESSES$P(N_{B})=0$ where $N_{B}:=\{\omega\in\Omega’ : Q^{\omega}(B)\neq\varphi_{B}(\omega)\}$ .

algebra which generates$\omega$$\mapsto \mathrm{Q}\mathrm{U}\mathrm{J}(\mathrm{B})$$\mathcal{F}$and $N:= \bigcup_{B\in A}N_{B}$ . Thenis I-measurable on

monotone class lemma, for$\Omega’\backslash N$$B\in \mathcal{F}$-invariant which writes also$\forall P\in \mathcal{L}_{\mathrm{s}\mathrm{t}}$Indeed, for any$f\in\tilde{\mathcal{F}}$,, $P(N)=0$ and, firstly for $B\in A$ , and then, thanks to the$P\in \mathcal{L}_{\mathrm{s}\mathrm{t}}$$Q^{\omega}\in \mathcal{L}_{\mathrm{s}\mathrm{t}}$$\forall f\in\overline{\mathcal{F}}$and any$\forall P\in \mathcal{L}_{\mathrm{s}\mathrm{t}}$.

(B) Now we prove that for any$T$

Let $A$ be acountable$\omega$in$\Omega’$,is$Q^{\omega}$. Firstly let us prove,$A\in \mathrm{I}$, for $P$-almost every$\mathrm{E}_{P}^{\mathrm{I}}(f)P- \mathrm{a}\mathrm{s}=.$.

$\mathrm{E}_{P}^{\mathrm{I}}(f\circ T)$

.

(5)

,

$\int_{\Omega}1_{A}(f\circ T)dP=\int_{\Omega}(1_{A}\circ T)(f\circ T)dP=\int_{\Omega}\cdot 1_{A}fd[T_{\#}P]=^{\theta}.\int_{\Omega}1_{A}fdP$.

Formula (5) can be written.

Now let $A$ be acountable algebra which generates$f=1_{B}$ ,one gets from (6)set $N$ such that
$P(\{\omega\in\Omega’ :$Q^{\omega}(B)P- \mathrm{a}\mathrm{s}=.$$\forall\omega\in\Omega’\backslash N$

Q^{\omega}\in \mathcal{L}_{\mathrm{s}\mathrm{t}}\})=1$(6)$\int_{\Omega}(f\circ T)dQ^{\omega}$$\int_{\Omega}fdQ^{\omega}=P- \mathrm{a}.\mathrm{s}$. $Q^{\omega}(T^{-1}B)$$\mathcal{F}$. Taking, for any $B\in A$ ,. So there exists aP-negligible, one has $[\forall B\in A, Q^{\omega}(B)=Q^{\omega}(T^{-1}B)]$ . Hence.

(C) Now we have to prove thatand has $P$-measure1for any$\Omega_{\mathrm{e}\mathrm{r}\mathrm{g}}:=$$P\in \mathcal{L}_{\mathrm{s}\mathrm{t}}${$\omega\in\Omega’$. Let for:$B\in \mathcal{F}$$Q^{\omega}$

is ergodic} belongs toand$\nu\in \mathcal{L}_{\mathrm{s}\mathrm{t}}$$f_{B}( \nu):=\int_{\Omega’}[Q^{\omega}(B)-\nu(B)]^{2}d\nu(\omega)$

Since $\nu(B)$ is the -mean of$\nu$$\omega\mapsto Q^{\omega}(B)$,$f_{B}( \nu):=\int_{\Omega’}Q^{\omega}(B)^{2}d\nu(\omega)-\nu(B)^{2}$.

,$\hat{\mathcal{F}}$46
M. VALADIERNow let us consider $Y_{B}(\omega):=Q^{\omega}(B)^{2}$ which is-measurable) in$\mathrm{Z}$$\omega$. Then, for$P\in \mathcal{L}_{\mathrm{s}\mathrm{t}}$,$f_{B}(Q^{\cdot})= \int_{\Omega}Y_{B}dQ^{\omega}-Y_{B}(\omega)=P- \mathrm{a}.\mathrm{s}$.

$[\mathrm{E}_{P}^{\mathrm{I}}(Y_{B})](\omega)-YB(\omega)=0P- \mathrm{a}.\mathrm{s}.$So there exists $N$ a -negligible set such that$P$

Now let us prove that if

and, taking

Q^{\omega}=\nu\})=1$$B\in \mathrm{I}$the implication,$\Leftarrow$

The functions$\forall B\in A$$\forall\omega\in\Omega’\backslash N$, $f_{B}(Q^{\omega})=0$ .

, [ is ergodic] $\Leftrightarrow[\forall B\in A, f_{B}(\nu)=0]$ . Firstly$\nu$. This ensures$1_{B}\nu- \mathrm{a}.\mathrm{s}=$.

$\forall B\in \mathcal{F}$. $\nu(B)$$\mathrm{E}_{\nu}^{\mathrm{I}}(1B)\nu- \mathrm{a}=.\mathrm{s}$,$\nu(\{(\mathrm{E}_{\nu}^{\mathrm{I}}(1B)) (.)=\nu(B)\})=1$

hence $\nu(B)=0$ or 1. This proves. The converse is easily checked.
are measurable functions with respect to the tribe on$f_{B}$

generated by the maps $P\mapsto P(C)$ (running through$C$$\mathcal{L}_{\mathrm{e}\mathrm{r}\mathrm{g}}=\cap\{\nu\in \mathcal{L}_{\mathrm{s}\mathrm{t}} :is ameasurable subset of$\Omega_{\mathrm{e}\mathrm{r}\mathrm{g}}\})=1$,.

implies $\nu(\{\omega\in\Omega’ : \forall B\in A, Q^{\omega}(B)=\nu(B)\})=1$ , hence$[\forall B\in A, f_{B}(\nu)=0]$$\nu(\{\omega\in\Omega’ :$\nu\in \mathrm{C}\mathrm{s}\mathrm{t}$-measurable (more precisely$\mathrm{J}$$\mathcal{L}_{\mathrm{s}\mathrm{t}}$and$\Omega_{\mathrm{e}\mathrm{r}\mathrm{g}}\in\hat{\mathcal{F}}$$\mathcal{F}$). Hencef_{B}(\nu)=0\}B\in A$. The property

follows from the foregoing observations.

$\mathcal{L}_{\mathrm{s}\mathrm{t}}$$P(\{\omega\in\Omega’$:$Q^{\omega}\in$$\square$Remarks. 1) There are two equivalence relations: firstly(R1)$Q^{\omega}=Q^{\omega’}$

which makes sense on$\Omega’$and secondly$\forall A\in \mathrm{I}$Let$\Gamma_{\omega}$

denote the class of

any $f\in W$ changing$\omega$in$T\omega$,$1_{A}(\omega)=1_{A}(\omega’)$$\omega\in\Omega’$

or in

.

(R2)

for (R1) and let us prove its invariance. For$T^{-1}\omega$does not change the Cesaro limit of the

47
STATIONARY PROCESSES

sequence$T^{-1}\omega$$(f(T^{j}\omega))_{j}$that isboth belong to$\Gamma_{\omega}$$\lim_{n}\int fdQ_{n}^{\omega}$, and since

The class of ci for (R2) isinvariant set containing$\omega$$T$$Q^{T\omega}=Q^{T^{-1}\omega}=Q^{\omega}$

is bijective,$\dot{\omega}=\{T^{j}\omega :j\in \mathbb{Z}\}$$T\Gamma_{\omega}=\Gamma_{\omega}$, so$T\omega$

and.

because this is the smallest Borel.

Thanks to the invariance of$\Gamma_{\omega}$

relation (R2) is always finer than (R1). But ingeneral they do not coincide neither on

set. Let us give an example: let

Dirac mass and. Hence$P:=\varpi^{\otimes \mathrm{Z}}$$\varpi$$\Omega_{\mathrm{e}\mathrm{r}\mathrm{g}}$nor on$\Omega\backslash N$

where $N$ is any negligible

be aprobability measure on $K$ not reduced to a

. Since the$X_{n}$are i.i.d.

$Q^{\omega}P- \mathrm{a}\mathrm{e}=.$. $P$(and as well known–[Do] Th.1.2 page 460 –the process is ergodic). So there is aunique class for(R1). For any $P$ negligible $N$ ,$\Omega\backslash N$

has the cardinal of$\mathbb{R}$(because $P$ is isomorphic

to the Lebesgue measure). The existence of only one class for (R2) would lead to acontradiction. Indeed, suppose there is only the class$\omega’$,$\exists j\in \mathbb{Z}$such that$\omega’=T^{j}\omega$

and the set$\Omega\backslash N$$\dot{\omega}=\{T^{j}\omega :j\in \mathbb{Z}\}$. For anyof trajectories under consideration

would be countable.

The set$\Omega\backslash \Omega_{\mathrm{e}\mathrm{r}\mathrm{g}}$is not the biggest $P$-negligible set possible. For adiscussion of

negligible sets when $K=\{0,1\}$ in connection with the notion of random numbers

see Dellacherie [De].
2) Since (R2) is finer than (R1),denotes its value on$\dot{\omega}$and$\dot{\Omega}$

is constant on each class$\dot{\omega}$; let$^{\dot{\omega}}$

the set of all classes. Then as aconsequence of (2), $P$

is the mixing of the ergodic laws$\omega\mapsto\dot{\omega}$$\omega\mapsto Q^{\omega}$$^{\dot{\omega}}$(ci$\in\dot{\Omega}$) according to the image of $P$ on$\dot{\Omega}$by. Thus (2) looks as (1) of Theorem 4. For historical works which attacked

disintegrating $P$ with respect to Isee Halmos [Hll] and Ambrose-Halmos-Kakutani[AHK].
3) Dynkin proves many other results: specially he gets (Theorem 3.1) that$\mathcal{L}_{\mathrm{s}\mathrm{t}}$48
M. VALADIER

is asimplex in the Choquet sense whose extreme points are the ergodic measures.

6. Further comments

When one observes only one trajectory of aprocess which is assumed to be station-ary, one can only identify the law$Q^{\omega}$corresponding to the observed trajectory. For

example if one observes aMarkov trajectory living in {1, 2} obeying to the tran-

sition matrix$(\begin{array}{ll}\frac{1}{4} \frac{1}{2}\frac{3}{4} \frac{1}{2}\end{array})$

then either the whole process is ergodic and obeys to thistransition matrix with probabilities of states equal to$(\begin{array}{l}\frac{2}{5}\frac{3}{5}\end{array})$or there exists other

ergodic classes about nothing is known.

Aremark about asmall strange phenomenon: suppose one observes$(x_{n})_{n\in}\mathrm{z}_{-}$

where $x_{n}=(-!)^{n}$ . One possibility is: there is not any random and this is justaperiodic behavior which may continue with $x_{n}=(-1)^{n}$ for $n\geq 1$ . If we aresure that there is behind astationary stochastic process then the observed trajectory continues in this way and the trajectory$(y_{n})_{n\in}\mathrm{z}$$=((-1)^{n+1})_{n\in \mathrm{Z}}$is another(hidden) possibility. So, if the process is ergodic, these two trajectories are theonly ones and have probability 1/2. This is the Markov chain with states {-1, 1},

matrix of transitions$(\begin{array}{ll}0 11 0\end{array})$and probabilities of states equal to$(\begin{array}{l}\frac{1}{2}\frac{1}{2}\end{array})$.

7. Prospects

Let us consider manufacturing of concrete. The dimensions and shapes of the stones

are random variables with stochastic characteristics which are the same as long as
the stones come from the same origin. This origin could change when building anew
work. This is contingency. But as long as the origin of stones remains unchanged,
all amounts as if the ergodic hypothesis was satisfied. To be more precise the result$\mathrm{s}$49
STATIONARY PROCESSES

of Section 5would have to be extended to the group$\mathbb{R}^{N}$in place of Z. Fomin [Fo]gives in Russian some results in this line; see also Remark 6.1 in [Dy, p.717]. Moregenerally the problem of relaxing the ergodic hypothesis into the stationarity one

comes up in stochastic homogenization. For afew references see Dal MasO-Modica[DMM1-2], Nguyen-Zessin and Licht-Michaille [LM1-2].

REFERENCES[AHK]

W. Ambrose, P.R. Halmos &S. Kakutani, The decomposition of measures, II, Duke

Math. J. 9(1942), 43-47.
[Ba]

H. Bauer, Probability theorry and elements of measure theory, Holt, Rinehart &Winston, New York, 1972 (translation of Wahrscheinlichkeitstheorie und Grundzuge der

Masstheorie, De Gruyter, Berlin, 1968).

[Bir]

G.D. Birkhoff, Proof of the ergodic theorem, Proc. Nat. Acad. Sci. U.S.A. 17 (1931),

656-660.
[BH]

J.R. Blum &D.L. Hanson, On invariant probability measures I, Pacific J. Math. 10(1960), 1125-1129.

[Bo]

N. Bourbaki, Integration, Chapitres 1\‘a 4, Hermann, Paris, 1965.

[Che]

F. Chersi, An ergodic decomposition of invariant measures for discrete semifiows on

standard Borel spaces, Advanced topics in the theory of dynamical systems. Proceedingsof the International Conference held in Trento, June 1-6, 1987 (G. Fusco, M. Iannelli&L. Salvadori, eds.), Notes and Reports in Mathematics in Science and Engineering,
vol. 6, 1989, PP.75-87.

[Chol]

G. Choquet, Existence et unicite des representations integrales au moyen des points
extr\’emaux dans les cones convexes, Seminaire Bourbaki, 9\‘eme annee (1956/1957), Paris,

1959, pp. 139.1-139.15.

[Ch02]

G. Choquet, Lectures on Analysis, vol. II Representation Theory, W.A. Benjamin, Inc.,New York, 1969.

50
M. VALADIER[Ch03]

G. Choquet, Reprisentation intigrale, Measure Theory and its Applications (J. Dubois&P. Morales,$\mathrm{e}\mathrm{d}\mathrm{s}.$$\mathrm{J}.\mathrm{M}$. Belley,), Lecture Notes in Mathematics, vol. 1033, Springer Verlag,

Berlin, 1983 Pp. 114-143.

[DMMI] G. Dal MasO&L. Modica, Nonlinear stochastic homogenization, Annali Mat. Pura ed
Appl. 144 (1986), 347-389.

[DMM2] G. Dal MasO&L. Modica, Nonlinear stochastic homogenization and ergodic theory, J.

Reine Angew. Math. 368 (1986), 28-42.
[De]

C. Dellacherie, Nombre au hasard de Borel \‘a Martin-Loef, Gazette des Mathematiciens(Soci\’et\’e Mathematique de France) 11 (1978), 23-58.

[DGS

M. Denker, C. Grillenberger&K. Sigmund, Ergodic theory on compact spaces, Lecture

Notes in Math., vol. 527, Springer-Verlag, Berlin, 1976.
[Do]$\mathrm{J}.\mathrm{L}$. Doob, Stochastic processes, Wiley, New York, 1965 (sixth printing. First printing,1953).

[Du]$\mathrm{R}.\mathrm{M}$. Dudley, Real analysis and probability, Wadsworth &Brooks/Cole Mathematics

Series, California, 1989.
[Dy]. Dynkin, Sufficient statistics and extreme points, Ann. Probability 6(1978), 705-$\mathrm{E}.\mathrm{B}$730.

. Farrell, Representation of invariant measures, Illinois J. Math. 6(1962), 447-467.

[Fa]$\mathrm{R}.\mathrm{H}$[Fi]

B. de Finetti, La privision :ses lois logiques, ses sources subjectives, Ann. Inst. H. Poin-care’ 7(1937), 1-68.

[Fo]$\mathrm{S}.\mathrm{V}$. Fomin, On measures invariant under certain groups of transformations, Izvestiya

Akad. Nauk SSSR Ser. Mat. 14 (1950), 261-274.
[Fu]

H. Furstenberg, Stationnary processes and prediction theory, Annals of Mathematics

Studies no44, Princeton University Press, Princeton, 1960 (MR (1963) 25 if 3573).
[Hll]$\mathrm{P}.\mathrm{R}$[H12]$\mathrm{P}.\mathrm{R}$[H13]$\mathrm{P}.\mathrm{R}$. Halmos, The decomposition of measures, Duke Math. J. 8(1941), 386-392.

. Halmos,$\Lambda f$easurable transformations, Bull. Amer. Math. Soc. 55 (1949), 1015-1034.

. Halmos, Lectures on Ergodic Theory, The Mathematical Society of Japan, Tokyo,

1956.

51
STATIONARY PROCESSES[HN]$\mathrm{P}.\mathrm{R}$. Halmos &J. von Neumann, Operator methods in classical mechanics $II$, Ann. of

Math. 43 (1942), 332-350.
[HS]

E. Hewitt &L.J. Savage, Syrnrnetric measures on cartesian products, Trans. Amer.
Math. Soc. 80 (1955), 470-501.

[HJ]

J. Hoffmann-Jorgensen, Eistence of conditional probabilities, Math. Scand. 28 (1971),

257-264.
[Ji]

M. Jirina, On regular conditional probabilities, Czechoslovak Math. J. 84 (1959), 445-

451.
[Ko]

$\mathrm{A}.\mathrm{N}$. Kolmogorov, Foundations of the theory of probability, Chelsea, New York, 1950

(translation of Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer, Berlin, 1933).

[KB]

N. Kryloff&N. Bogoliouboff, La thiorie generale de la mesure dans son application \‘aI’etude des systemes dynamiques de la mecanique non lineaire, Ann. of Math. 38 (1937),65-113.
[LM1]

G. Licht &G. Michaille, Convergence results for subadditive processes, Prepublication

no28 (15 pages) (1994), Depart, de Math., Universite’ de Montpellier, France.
[LM2]

G. Licht &G. Michaille, Global-local subadditive ergodic theorems and applications tohomogenization in elasticity, submitted (1999).

[Mah]

D. Maharam, Decompositions of rneaserre algebras and spaces, Trans. Amer. Math. Soc.

69 (1950), 142-160.
[N1]

J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. of Math.
33 (1932), 587-642, 789-791 (pages 307-365 in [N4]).

[N2]

J. von Neumann, Proof of the quasi-ergodic hypothesis, Proc. Nat. Acad. Sci. U.S.A. 18(1932), 70-82 (pages 261-273 in [N4]).

[N3]

J. von Neumann, Physical applications of the ergodic hypothesis, Proc. Nat. Acad. Sci.

U.S.A. 18 (1932), 263-266 (pages 274-277 in [N4]).
[N4]

J. von Neumann, Collected works, vol.
functions in agroup ([NZ]$\mathrm{X}.\mathrm{X}$$\mathrm{A}.\mathrm{H}$$\mathrm{I}\mathrm{I}$, Operators, Ergodic theory and almost periodic. Taub, ed.), Pergamon Press, Oxford, 1961.

. Nguyen&H. Zessin, Ergodic theorems for spatial processes, Z. Wahrscheinlichkeit

52
M. VALADIER

stheorie verw. Gebiete 48 (1979), 133-158.

. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116-136.

[Ox]$\mathrm{J}.\mathrm{C}$[Ph]$\mathrm{R}.\mathrm{R}$[RR]

S. Ramakrishnan&B.V. Rao,

. Phelps, Lectures on Choquet’s theorem, Van Nostrand, Princeton, 1966.
$B$-spaces are standard Borel, The Annals of Probability8(1980), 1191.

[T]

M. Talagrand, Capacitis invariantes extremales, Ann. Inst. Fourier 28 (1978), 79-146.

[V1]

IVI. Valadier, Ddsint\’egmtion$d$’une mesure sur un produit, C. R. Acad. Sci. Paris Sir.

A276 (1973), 33-35 (summary of Comparaison de trois thioremes de disintigration,

SUm. Anal. Convexe 2(1972) 10.1-10.21).
M. Valadier, Structure des atomes conditionnels

[V2]

$d$

’un espace de probabiliti, Sem. Anal.

Convexe 10 (1980), 1.1-1. 17.
[珂 ]$\mathrm{V}.\mathrm{S}$$\mathrm{a}\mathrm{r}$. Varadarajan, Groups of automorphisms of Borel spaces, Trans. Amer. Math. Soc.

109 (1963), 191-220.

$\mathrm{D}\acute{\mathrm{B}}\mathrm{P}\mathrm{A}\kappa \mathrm{r}\mathrm{B}\mathrm{M}\mathrm{B}\mathrm{N}\mathrm{T}\mathrm{D}\mathrm{B}\mathrm{M}\mathrm{A}\mathrm{T}\mathrm{H}\acute{\mathrm{B}}\mathrm{M}\mathrm{A}\mathrm{T}1\mathrm{Q}\mathrm{U}\mathrm{B}\mathrm{S}-\mathrm{C}\mathrm{A}\mathrm{S}\mathrm{B}051$, UNIVBRSITE MONTPELLIERBATAILLON, 34095 MONTPELLIER CEDEX 5FRANCE-mail address: valadier@math.univ-m0ntp2.fr$E$$\mathrm{I}\mathrm{I}$, place EUG\‘ENE