User:Boris Tsirelson/sandbox1

Measure algebra may refer to:

• algebra of measures on a topological group with the operation of convolution; see measure algebra (harmonic analysis);
• normed Boolean algebra, either in general or consisting of equivalence classes of measurable sets; see measure algebra (measure theory).

Measure algebra (measure theory)

Category:Classical measure theory

$\newcommand{\Om}{\Omega} \newcommand{\om}{\omega} \newcommand{\F}{\mathcal F} \newcommand{\B}{\mathcal B} \newcommand{\M}{\mathcal M} $ A measure algebra is a pair $(B,\mu)$ where $B$ is a Boolean σ-algebra and $\mu$ is a (strictly) positive measure on $B$. However, about the greatest value $\mu(\bsone_B)$ of $\mu$, assumptions differ from $\mu(\bsone_B)=1$ (that is, $\mu$ is a probability measure) in [Ha2, p. 43] and [K, Sect. 17.F] to $\mu(\bsone_B)<\infty$ (that is, $\mu$ is a totally finite measure) in [G, Sect. 2.1] to $\mu(\bsone_B)\le\infty$ in [P, Sect. 1.4C] and [Ha1, Sect. 40].

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Also: Lebesgue-Rokhlin space

A probability space is called standard if it satisfies the following equivalent conditions:

• it is almost isomorphic to the real line with some probability distribution (in other words, a completed Borel probability measure, that is, a Lebesgue–Stieltjes probability 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.

References

[P]	Karl Petersen, "Ergodic theory", Cambridge (1983).   MR0833286   Zbl 0507.28010
[H1]	P.R. Halmos, "Measure theory", Van Nostrand (1950).   MR0033869   Zbl 0040.16802
[H2]	P.R. Halmos, "Lectures on ergodic theory", Math. Soc. Japan (1956).   MR0097489   Zbl 0073.09302
[G]	Eli Glasner, "Ergodic theory via joinings", Amer. Math. Soc. (2003).   MR1958753   Zbl 1038.37002
[K]	Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995).   MR1321597   Zbl 0819.04002
[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
[F]	D.H. Fremlin, "Measure theory", Torres Fremlin, Colchester. Vol. 1: 2004   MR2462519   Zbl 1162.28001; Vol. 2: 2003   MR2462280   Zbl 1165.28001; Vol. 3: 2004   MR2459668   Zbl 1165.28002; Vol. 4: 2006   MR2462372   Zbl 1166.28001