Difference between revisions of "User:Boris Tsirelson/sandbox1"
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\newcommand{\M}{\mathcal M} $ | \newcommand{\M}{\mathcal M} $ | ||
A [[probability space]] is called '''standard''' if it satisfies the following equivalent conditions: | A [[probability space]] is called '''standard''' if it satisfies the following equivalent conditions: | ||
− | * it is almost isomorphic to the real line with some [[Lebesgue–Stieltjes integral| | + | * it is [[Measure space#isomorphism|almost isomorphic]] to the real line with some [[Lebesgue–Stieltjes integral| |
Lebesgue–Stieltjes measure]]; | Lebesgue–Stieltjes measure]]; | ||
− | * it is a [[standard Borel space]] endowed with a [[probability measure]], completed, and possibly augmented with a null set; | + | * it is a [[standard Borel space]] endowed with a [[probability measure]], [[Measure space#completion|completed]], and possibly augmented with a [[Measure space#null|null set]]; |
− | * it is complete, [[Measure space#perfect|perfect]], and the corresponding Hilbert space is separable. | + | * it is [[Measure space#completion|complete]], [[Measure space#perfect|perfect]], and the corresponding Hilbert space is separable. |
Revision as of 10:07, 29 February 2012
$\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 Lebesgue–Stieltjes 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.
(See Definition 1 below.) Every standard probability space is isomorphic (mod 0) to an interval with Lebesgue measure, a finite or countable set of atoms, or a combination of both. (See Theorem ? below.)
Example. The set of all continuous functions $[0,\infty)\to\R$ with the Wiener measure is a standard probability space.
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 |
[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 |
Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=21351