Difference between revisions of "User:Boris Tsirelson/sandbox1"
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| − | A '''measure algebra''' is a pair $(B,\mu)$ where $B$ is a Boolean σ-algebra and $\mu$ is a (strictly) positive measure on $B$. The (strict) positivity means $\mu(x)\ge0$ and $\mu(x)=0\iff x=\bszero_B$ for all $x\in B$. 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 {{Cite| | + | A '''measure algebra''' is a pair $(B,\mu)$ where $B$ is a Boolean σ-algebra and $\mu$ is a (strictly) positive measure on $B$. The (strict) positivity means $\mu(x)\ge0$ and $\mu(x)=0\iff x=\bszero_B$ for all $x\in B$. 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 {{Cite|H2|p. 43}} and {{Cite|K|Sect. 17.F}} to $\mu(\bsone_B)<\infty$ (that is, $\mu$ is a totally finite measure) in {{Cite|G|Sect. 2.1}} to $\mu(\bsone_B)\le\infty$ in {{Cite|P|Sect. 1.4C}} and {{Cite|H1|Sect. 40}}. |
| + | A ''measure algebra of a measure space'' $(X,\A,\mu)$ consists, by definition, of all equivalence classes of measurable sets. (Sets of $\A$ or its $\mu$-completion $\A_\mu$ give the same result.) | ||
| + | |||
| + | This is "a coherent way to ignore the sets of measure $0$ in a measure space" {{Cite|P|page 15}}. "Many of the difficulties of measure theory and all the pathology of the subject arise from the existence of sets of measure zero. The algebraic treatment gets rid of this source of unpleasantness by refusing to consider sets at all; it considers sets modulo sets of measure zero instead." {{Cite|H2|page 42}} | ||
Revision as of 09:36, 16 March 2012
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{\A}{\mathcal A} \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 $B$. The (strict) positivity means $\mu(x)\ge0$ and $\mu(x)=0\iff x=\bszero_B$ for all $x\in B$. 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 [H2, 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 [H1, Sect. 40].
A measure algebra of a measure space $(X,\A,\mu)$ consists, by definition, of all equivalence classes of measurable sets. (Sets of $\A$ or its $\mu$-completion $\A_\mu$ give the same result.)
This is "a coherent way to ignore the sets of measure $0$ in a measure space" [P, page 15]. "Many of the difficulties of measure theory and all the pathology of the subject arise from the existence of sets of measure zero. The algebraic treatment gets rid of this source of unpleasantness by refusing to consider sets at all; it considers sets modulo sets of measure zero instead." [H2, page 42]
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 |
| [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 |
| [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 |
Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=21688