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| − | ====On terminology==== | + | ====Criticism==== |
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| − | The term "standard probability space" is used in {{Cite|I}}. The same, or very similar, notion appears also as: "Lebesgue space" {{Cite|Ro}}, {{Cite|Ru}}, {{Cite|P}}, {{Cite|G}}; "standard Lebesgue space" {{Cite|G}}; "Lebesgue-Rohlin space" {{Cite|H}}, {{Cite|B}}; and "L. R. space" {{Cite|H}}.
| + | A quote from {{Cite|Dur|Sect. 1.4(c), p. 33}}: |
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| | + | : $(S,\mathcal S)$ is said to be ''nice'' if there is a 1-1 map $\phi$ from $S$ into $\R$ so that $\phi$ and $\phi^{-1}$ are both measurable. |
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| | + | : Such spaces are often called ''standard Borel spaces,'' but we already have too many things named after Borel. The next result shows that most spaces arising in applications are nice. |
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| | + | : (4.12) ''Theorem.'' If $S$ is a Borel subset of a complete separable metric space $M$, and $\mathcal S$ is the collection of Borel subsets of $S$, then $(S,\mathcal S)$ is nice. |
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| − | Some authors admit totally finite (not necessarily probability) measures {{Cite|P}}, {{Cite|B}}. Note also "standard σ-finite measure" in {{Cite|Mac}}. Some authors exclude spaces of cardinality higher than continuum ({{Cite|Ro}}, {{Cite|Ru}}, {{Cite|G}}, but not {{Cite|I}}, {{Cite|H}}, {{Cite|Mac}}, {{Cite|P}}, {{Cite|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 {{Cite|B}}, {{Cite|G}}.
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| − | ====Criticism====
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| − | According to {{Cite|Mal}},
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| − | * a measure is called separable if the corresponding $L_1$ space is separable {{Cite|Mal|Sect. IV.6.0}};
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| − | * every separable complete nonatomic probability space is isomorphic to $[0,1]$ with Lebesgue measure {{Cite|Mal|Sect. IV.6.4.2: "structure theorem (nonatomic case)"}}.
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| − | 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 {{Cite|KP|p. 94}}, {{Cite|D|p. 1002}}).
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| − | Further, in {{Cite|Mal|Sect. IV.6.4.3: "structure theorem (general case)"}} it is claimed that every separable (as defined there) complete probability space is standard (as defined here), which is wrong, of course.
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| | ====References==== | | ====References==== |
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| | |valign="top"|{{Ref|I}}|| Kiyosi Itô, "Introduction to probability theory", Cambridge (1984). {{MR|0777504}} {{ZBL|0545.60001}} | | |valign="top"|{{Ref|I}}|| Kiyosi Itô, "Introduction to probability theory", Cambridge (1984). {{MR|0777504}} {{ZBL|0545.60001}} |
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| − | |valign="top"|{{Ref|Ru}}|| Thierry de la Rue, "Espaces de Lebesgue", ''Séminaire de Probabilités XXVII,'' Lecture Notes in Mathematics, '''1557''' (1993), Springer, Berlin, pp. 15–21. {{MR|1308547}} {{ZBL|0788.60001}}
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| − | |valign="top"|{{Ref|H}}|| Jean Haezendonck, "Abstract Lebesgue-Rohlin spaces", ''Bull. Soc. Math. de Belgique'' '''25''' (1973), 243–258. {{MR|0335733}} {{ZBL|0308.60006}}
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| − | |valign="top"|{{Ref|HN}}|| P.R. Halmos, J. von Neumann, "Operator methods in classical mechanics, II", ''Annals of Mathematics (2)'' '''43''' (1942), 332–350. {{MR|0006617}} {{ZBL|0063.01888}}
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| − | |valign="top"|{{Ref|Ro}}|| V.A. Rokhlin, (1962), "On the fundamental ideas of measure theory", ''Translations (American Mathematical Society) Series 1,'' 10 (1962), 1–54. {{MR|0047744}} Translated from Russian: Рохлин, В. А. (1949), "Об основных понятиях теории меры", Математический Сборник (Новая Серия) 25(67): 107–150. {{MR|0030584}}
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| − | |valign="top"|{{Ref|P}}|| Karl Petersen, "Ergodic theory", Cambridge (1983). {{MR|0833286}} {{ZBL|0507.28010}}
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| − | |valign="top"|{{Ref|G}}|| Eli Glasner, "Ergodic theory via joinings", Amer. Math. Soc. (2003). {{MR|1958753}} {{ZBL|1038.37002}}
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| − | |valign="top"|{{Ref|B}}|| V.I. Bogachev, "Measure theory", Springer-Verlag (2007). {{MR|2267655}} {{ZBL|1120.28001}}
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| − | |valign="top"|{{Ref|Mac}}|| George W. Mackey, "Borel structure in groups and their duals", ''Trans. Amer. Math. Soc.'' '''85''' (1957), 134–165. {{MR|0089999}} {{ZBL|0082.11201}}
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| − | |valign="top"|{{Ref|Mal}}|| Paul Malliavin, "Integration and probability", Springer-Verlag (1995). {{MR|1335234}} {{ZBL|0874.28001}}
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| − | |valign="top"|{{Ref|KP}}|| Joseph Kupka, Karel Prikry, "The measurability of uncountable unions", ''Amer. Math. Monthly'' '''91''' (1984), 85–97. {{MR|0729548}} {{ZBL|0533.28010}}
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| − | |valign="top"|{{Ref|D}}|| R.M. Dudley, "Nonmetric compact spaces and nonmeasurable processes", ''Proc. Amer. Math. Soc.'' '''108''' (1990), 1001–1005. {{MR|0994775}} {{ZBL|0697.60037}}
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