User:Boris Tsirelson/sandbox1

$\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 (be they absolutely continuous or singular), 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, connectedness, compactness etc. 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 probability-space 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, which also has a probability-space counterpart.

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 [R, Th. 3-2] and [H, Prop. 9].)

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Non-example. The set $[0,1]^\R$ of all functions $\R\to[0,1]$ with the product of Lebesgue measures is a nonstandard probability space.

Definition 1a. A probability space $(\Om,\F,P)$ is standard if it is complete and there exist a subset $\Om_1\subset\Om$ and a σ-field (in other words, σ-algebra) $\B$ on $\Om_1$ such that $(\Om_1,\B)$ is a standard Borel space and every set of $\F$ is almost equal to a set of $\B$. (See [I, Sect. 2.4].) (Clearly, $\Om_1$ must be of full measure.)

Definition 1b (equivalent). A probability space $(\Om,\F,P)$ is standard if it is complete, perfect and countably separated mod 0 in the following sense: some subset of full measure, treated as a subspace of the measurable space $(\Om,\F)$, is a countably separated measurable space.

(See [I, Sect. 3.1] for a proof of equivalence of these definitions.)

On terminology

Also "Lebesgue-Rokhlin space" and "Lebesgue space".

In [M, Sect. 6] universally measurable spaces are called metrically standard Borel spaces.

In [K, Sect. 21.D] universally measurable subsets of a standard (rather than arbitrary) measurable space are defined.

In [N, Sect. 1.1] an absolute measurable space is defined as a separable metrizable topological space such that every its homeomorphic image in every such space (with the Borel σ-algebra) is a universally measurable subset. The corresponding measurable space (with the Borel σ-algebra) is also called an absolute measurable space in [N, Sect. B.2].

References

[I]	Kiyosi Itô, "Introduction to probability theory", Cambridge (1984).   MR0777504   Zbl 0545.60001
[R]	Thierry de la Rue, "Espaces de Lebesgue", Séminaire de Probabilités XXVII, Lecture Notes in Mathematics, 1557 (1993), Springer, Berlin, pp. 15–21.    Zbl 0788.60001
[B]	V.I. Bogachev, "Measure theory", Springer-Verlag (2007).   MR2267655  Zbl 1120.28001
[C]	Donald L. Cohn, "Measure theory", Birkhäuser (1993).   MR1454121   Zbl 0860.28001
[D]	Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989).   MR0982264   Zbl 0686.60001
[M]	George W. Mackey, "Borel structure in groups and their duals", Trans. Amer. Math. Soc. 85 (1957), 134–165.   MR0089999   Zbl 0082.11201
[K]	Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995).   MR1321597   Zbl 0819.04002
[N]	Togo Nishiura, "Absolute measurable spaces", Cambridge (2008).   MR2426721   Zbl 1151.54001