User:Boris Tsirelson/sandbox1

On terminology

The term "standard probability space" is used in [I]. The same, or very similar, notion appears also as: "Lebesgue space" [Ro], [Ru], [P], [G]; "standard Lebesgue space" [G]; "Lebesgue-Rohlin space" [H], [B]; and "L. R. space" [H].

Some authors admit totally finite (not necessarily probability) measures [P], [B]. Note also "standard σ-finite measure" in [Mac]. Some authors exclude spaces of cardinality higher than continuum ([Ro], [Ru], [G], but not [I], [H], [Mac], [P], [B]) even though such space can be almost isomorphic to $(0,1)$ with Lebesgue measure (since it can contain a null set of arbitrary cardinality). Also, some authors do not insist on completeness [B], [G].

Criticism

According to [Mal],

• a measure is called separable if the corresponding $L_1$ space is separable [Mal, Sect. IV.6.0];
• every separable complete nonatomic probability space is isomorphic to $[0,1]$ with Lebesgue measure [Mal, Sect. IV.6.4.2: "structure theorem (nonatomic case)"].

The proof provides a measure preserving map from the given space to $[0,1]$ that generates the given σ-algebra. However, such map is not necessarily an isomorphism. Its image must be of full outer measure, but not of full inner measure, which is a manifestation of the "image measure catastrophe" (see [KP, p. 94], [D, p. 1002]).

Further, in [Mal, Sect. IV.6.4.3: "structure theorem (general case)"] it is claimed that every separable (as defined there) complete nonatomic probability space is standard (as defined here), which is wrong, of course.

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
[P]	Karl Petersen, "Ergodic theory", Cambridge (1983).   MR0833286   Zbl 0507.28010
[G]	Eli Glasner, "Ergodic theory via joinings", Amer. Math. Soc. (2003).   MR1958753   Zbl 1038.37002
[B]	V.I. Bogachev, "Measure theory", Springer-Verlag (2007).   MR2267655  Zbl 1120.28001
[Mac]	George W. Mackey, "Borel structure in groups and their duals", Trans. Amer. Math. Soc. 85 (1957), 134–165.   MR0089999   Zbl 0082.11201
[Mal]	Paul Malliavin, "Integration and probability", Springer-Verlag (1995).   MR1335234   Zbl 0874.28001