User:Boris Tsirelson/sandbox1

Measure algebra may refer to:

• algebra of measures on a topological group with the operation of convolution; see measure algebra (harmonic analysis);
• normed Boolean algebra, either in general or consisting of equivalence classes of measurable sets; see measure algebra (measure theory).

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Also: Lebesgue-Rokhlin space

Category:Classical measure theory

$\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 (be they 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].)

References

[P]	Karl Petersen, "Ergodic theory", Cambridge (1983).   MR0833286   Zbl 0507.28010
[H1]	P.R. Halmos, "Measure theory", Van Nostrand (1950).   MR0033869   Zbl 0040.16802
[H2]	P.R. Halmos, "Lectures on ergodic theory", Math. Soc. Japan (1956).   MR0097489   Zbl 0073.09302
[G]	Eli Glasner, "Ergodic theory via joinings", Amer. Math. Soc. (2003).   MR1958753   Zbl 1038.37002
[K]	Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995).   MR1321597   Zbl 0819.04002
[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
[F]	D.H. Fremlin, "Measure theory", Torres Fremlin, Colchester. Vol. 1: 2004   MR2462519   Zbl 1162.28001; Vol. 2: 2003   MR2462280   Zbl 1165.28001; Vol. 3: 2004   MR2459668   Zbl 1165.28002; Vol. 4: 2006   MR2462372   Zbl 1166.28001