Standard probability space

Also: Lebesgue-Rokhlin space

2020 Mathematics Subject Classification: Primary: 28Axx Secondary: 28A5060A10 [MSN][ZBL]

$\newcommand{\Om}{\Omega} \newcommand{\om}{\omega} \newcommand{\F}{\mathcal F} \newcommand{\B}{\mathcal B} \newcommand{\M}{\mathcal M} $ A probability space is called standard if it satisfies the following equivalent conditions:

• it is almost isomorphic to the real line with some probability distribution (in other words, a completed Borel probability measure, that is, a Lebesgue–Stieltjes probability measure);
• it is a standard Borel space endowed with a probability measure, completed, and possibly augmented with a null set;
• it is complete, perfect, and the corresponding Hilbert space is separable.

The isomorphism theorem

Every standard probability space consists of an atomic (discrete) part and an atomless (continuous) part (each part may be empty). The discrete part is finite or countable; here, all subsets are measurable, and the probability of each subset is the sum of probabilities of its elements.

Theorem 1. All atomless standard probability spaces are mutually almost isomorphic.

That is, up to almost isomorphism we have "the" atomless standard probability space. Its "incarnations" include the spaces $\R^n$ with atomless probability distributions (absolutely continuous, singular or mixed), as well as the set of all continuous functions $[0,\infty)\to\R$ with the Wiener measure. That is instructive: topological notions such as dimension do not apply to probability spaces.

Measure preserving maps

The inverse to a bijective measure preserving map is measure preserving provided that it is measurable; in this (not general) case the given map is a strict isomorphism. Here is an important fact in two equivalent forms.

Theorem 2a. Every bijective measure preserving map between standard probability spaces is a strict isomorphism.

Theorem 2b. If $(\Om,\F,P)$ is a standard probability space and $\F_1\subset\F$ a sub-σ-field such that $(\Om,\F_1,P|_{\F_1})$ is also standard then $\F_1=\F$.

Recall a topological fact similar to Theorem 2: if a bijective map between compact Hausdorff topological spaces is continuous then it is a homeomorphism. Moreover, if a Hausdorff topology is weaker than a compact topology then these two topologies are equal, which has the following measure-theory counterpart stronger than Theorem 2 (in two equivalent forms). Here we call a probability space countably separated if its underlying measurable space is countably separated.

Theorem 3a. Every bijective measure preserving map from a standard probability space to a countably separated complete probability space is a strict isomorphism.

Theorem 3b. If $(\Om,\F,P)$ is a standard probability space and $\F_1\subset\F$ is a countably separated sub-σ-field then $(\Om,\F,P)$ is the completion of $(\Om,\F_1,P|_{\F_1})$.

A continuous image of a compact topological space is always a compact set. In contrast, the image of a measurable set under a (non-bijective) measure-preserving map need not be measurable (indeed, the image of a null set need not be null; try the projection $\R^2\to\R^1$). Nevertheless, Theorem 4 (below) is a partial measure-theory counterpart, stronger than Theorem 3.

Theorem 4. Let $(\Om,\F,P)$ be a standard probability space, $(\Om_1,\F_1,P_1)$ a countably separated complete probability space, and $f:\Om\to\Om_1$ a measure preserving map. Then $(\Om_1,\F_1,P_1)$ is also standard, and $A_1\in\F_1\iff A\in\F$ whenever $A_1\subset\Om_1$ and $A=f^{-1}(A_1)$. In particular, the image $f(\Om)$belongs to $\F_1$. (See [Ru, Th. 3-2] and [H, Prop. 9].)

Quotient spaces

Theorem 4 (above) will be combined with the bijective correspondence between sub-σ-fields and linear sublattices described in the corresponding section of "Measure space". Here, as well as there, we restrict ourselves to σ-fields that contain all null sets.

Every measure preserving map $\alpha:\Om\to\Om'$ between standard probability spaces $(\Om,\F,P)$ and $(\Om',\F',P')$ leads to an embedding $f\mapsto f\circ\alpha$ of Hilbert spaces, $L_2(\Om',\F',P')\to L_2(\Om,\F,P)$. It is, moreover, an embedding of linear lattices, and therefore $L_2(\Om',\F',P')=L_2(\Om,\F_1,P|_{\F_1})$ (both embedded into $L_2(\Om,\F,P)$) for some sub-σ-field $\F_1\subset\F$. Clearly, $\F_1$ is generated by $\alpha$ (up to the null sets), and we may say that $(\Om',\F',P')$ is the quotient space of $(\Om,\F,P)$ by $\F_1$ (via $\alpha$).

Existence. Let $(\Om,\F,P)$ be a standard probability space and $\F_1\subset\F$ a sub-σ-field; then $\F_1$ is generated by some $\alpha$ (as above), which means existence of a quotient space of $(\Om,\F,P)$ by $\F_1$. Here is how to do it. Using separability of $L_2(\Om,\F_1,P|_{\F_1})$ one constructs a measurable map $\alpha:\Om\to\Om'$ from $(\Om,\F,P)$ to a standard measurable space $(\Om',\B)$ such that a function of $L_2(\Om,\F,P)$ belongs to $L_2(\Om,\F_1,P|_{\F_1})$ if and only if it is of the form $g\circ\alpha$ for some measurable $g:\Om'\to\R$. Taking the image of the measure $P$ under $\alpha$ and applying Theorem 4 one gets a standard probability space $(\Om',\F',P')$ and a measure preserving map $\alpha:\Om\to\Om'$ that generates $\F_1$.

Uniqueness. If also $(\Om'',\F'',P'')$ is the quotient space of $(\Om,\F,P)$ by $\F_1$ (via $\beta$) then there exists an almost isomorphism $\gamma$ from $(\Om',\F',P')$ to $(\Om'',\F'',P'')$ such that $\gamma\circ\alpha=\beta$, which means uniqueness of the quotient space up to almost isomorphism.

Existence of $\gamma$ (above) follows from the following fact. Let $(\Om,\F,P)$, $(\Om',\F',P')$ and $(\Om'',\F'',P'')$ be standard probability spaces, and $\alpha:\Om\to\Om'$, $\beta:\Om\to\Om''$ measure preserving maps. If the sub-σ-field generated by $\beta$ is contained in the sub-σ-field generated by $\alpha$ then $\beta=\gamma\circ\alpha$ for some (almost unique) measure preserving map $\gamma:\Om'\to\Om''$. This is basically the Doob-Dynkin lemma.

Let $(\Om,\F,P)$ be a standard probability space, $\F_1,\F_2\subset\F$ two independent sub-σ-fields, and $(\Om',\F',P')$, $(\Om'',\F'',P'')$ the corresponding quotient spaces (via $\alpha$, $\beta$); then the product space $(\Om',\F',P')\times(\Om'',\F'',P'')$ is the quotient space of $(\Om,\F,P)$ by $\sigma(\F_1,\F_2)$ (via $\alpha\times\beta:\omega\mapsto(\alpha(\omega),\beta(\omega))$). Here $\sigma(\F_1,\F_2)$ is the sub-σ-field generated by $\F_1,\F_2$. If, in addition, $\sigma(\F_1,\F_2)=\F$ then $\alpha\times\beta$ is an almost isomorphism from $(\Om,\F,P)$ to $(\Om',\F',P')\times(\Om'',\F'',P'')$. In this sense, any two independent sub-σ-fields $\F_1,\F_2$ that generate $\F$ decompose $(\Om,\F,P)$ into the product of two standard probability spaces (quotient spaces). The same holds for any finite or countable family of independent sub-σ-fields.

Conditional measures

Let $\alpha:\Om\to\Om'$ be a measure preserving map between standard probability spaces $(\Om,\F,P)$ and $(\Om',\F',P')$.

Theorem 5a (existence). There exist families $(\F_{\om'})_{\om'\in\Om'}$, $(P_{\om'})_{\om'\in\Om'}$ of σ-fields $\F_{\om'}$ on $\Om$ and probability measures $P_{\om'}$ on $(\Om,\F_{\om'})$ such that for almost every $\om'\in\Om'$

• $(\Om,\F_{\om'},P_{\om'})$ is a standard probability space,
• $\alpha(\om)=\om'$ for $P_{\om'}$-almost all $\om\in\Om$,

and for every $A\in\F$

• the function $\om'\mapsto P_{\om'}(A)$ on $(\Om',\F',P')$ is measurable,
• $P(A)=\int_{\Om'} P_{\om'}(A)\,P'(\!\rd\om')$.

Theorem 5b (uniqueness). If also families $(\F'_{\om'})_{\om'\in\Om'}$, $(P'_{\om'})_{\om'\in\Om'}$ satisfy the requirements of Theorem 5a then $\F_{\om'}=\F'_{\om'}$ and $P_{\om'}=P'_{\om'}$ for almost all $\om'\in\Om'$.

The measure $P_{\om'}$ is called the conditional measure on the subset $\{\om:\alpha(\om)=\om'\}$ of $\Om$, or the conditional distribution of $\om$ given $\alpha(\om)=\om'$.

Example. The projection $(x,y)\mapsto x$ from the square $(0,1)\times(0,1)$ with the two-dimensional Lebesgue measure to the interval $(0,1)$ with the one-dimensional Lebesgue measure is a measure preserving map. The conditional distribution of $(x,y)$ given $x$ is the one-dimensional Lebesgue measure on the interval $\{x\}\times(0,1)$ with the one-dimensional Lebesgue measure. This example is trivial, but note the different σ-fields: neither $\F_{\om'}\subset\F$ nor $\F\subset\F_{\om'}$.

On history

The notion (but not the term) "standard probability space" and Theorem 1 (isomorphism) were published by Halmos and von Neumann in 1942 [HN] and by Rokhlin in 1949 [Ro] following Rokhlin's unpublished manuscript of 1940 (according to [Ro, p. 2]). Theorem 5 (conditional measures) is due to Rokhlin [Ro, Sect. 3.1].

References

[I]	Kiyosi Itô, "Introduction to probability theory", Cambridge (1984).   MR0777504   Zbl 0545.60001
[Ru]	Thierry de la Rue, "Espaces de Lebesgue", Séminaire de Probabilités XXVII, Lecture Notes in Mathematics, 1557 (1993), Springer, Berlin, pp. 15–21.   MR1308547   Zbl 0788.60001
[H]	Jean Haezendonck, "Abstract Lebesgue-Rohlin spaces", Bull. Soc. Math. de Belgique 25 (1973), 243–258.   MR0335733   Zbl 0308.60006
[HN]	P.R. Halmos, J. von Neumann, "Operator methods in classical mechanics, II", Annals of Mathematics (2) 43 (1942), 332–350.   MR0006617   Zbl 0063.01888
[Ro]	V.A. Rokhlin, (1962), "On the fundamental ideas of measure theory", Translations (American Mathematical Society) Series 1, 10 (1962), 1–54.   MR0047744   Translated from Russian: Рохлин, В. А. (1949), "Об основных понятиях теории меры", Математический Сборник (Новая Серия) 25(67): 107–150.   MR0030584