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 $B$. 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 [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].

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