Difference between revisions of "User:Boris Tsirelson/sandbox1"
From Encyclopedia of Mathematics
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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}}. | 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}}. | ||
| − | 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. | + | 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}}. |
====References==== | ====References==== | ||
Revision as of 06:41, 6 April 2012
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].
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 |
How to Cite This Entry:
Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=24231
Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=24231