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 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