|
|
| (162 intermediate revisions by the same user not shown) |
| Line 1: |
Line 1: |
| − | '''Measure algebra''' may refer to:
| + | <ref> [http://hea-www.harvard.edu/AstroStat http://hea-www.harvard.edu/AstroStat]; <nowiki> http://www.incagroup.org </nowiki>; <nowiki> http://astrostatistics.psu.edu </nowiki> </ref> |
| | | | |
| − | * algebra of measures on a topological group with the operation of convolution; see [[measure algebra (harmonic analysis)]];
| + | ====Notes==== |
| − | * normed Boolean algebra, either in general or consisting of equivalence classes of measurable sets; see [[measure algebra (measure theory)]].
| + | <references /> |
| | | | |
| − | =Measure algebra (measure theory)=
| + | ------------------------------------------- |
| | | | |
| − | {{MSC.|28Axx|28A50,60A10}}
| |
| | | | |
| − | [[:Category:Classical measure theory]]
| + | {| |
| | + | | A || B || C |
| | + | |- |
| | + | | X || Y || Z |
| | + | |} |
| | | | |
| − | {{TEX|done}}
| |
| | | | |
| − | $\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 {{Cite|Ha2|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|Ha1|Sect. 40}}.
| |
| | | | |
| | + | ----------------------------------------- |
| | + | ----------------------------------------- |
| | | | |
| − | -------------------------------
| + | $\newcommand*{\longhookrightarrow}{\lhook\joinrel\relbar\joinrel\rightarrow}$ |
| − | ''Also: Lebesgue-Rokhlin space''
| |
| | | | |
| | + | <asy> |
| | + | size(100,100); |
| | + | label(scale(1.7)*'$T(\\Sigma)\hookrightarrow T(\\Sigma,X)$',(0,0)); |
| | + | </asy> |
| | | | |
| − | A [[probability space]] is called '''standard''' if it satisfies the following equivalent conditions:
| + | <asy> |
| − | * it is [[Measure space#Isomorphism|almost isomorphic]] to the real line with some [[probability distribution]] (in other words, a [[Measure space#Completion|completed]] [[Borel measure|Borel]] [[probability measure]], that is, a [[Lebesgue–Stieltjes integral|Lebesgue–Stieltjes]] probability measure);
| + | size(220,220); |
| − | * it is a [[standard Borel space]] endowed with a [[probability measure]], completed, and possibly augmented with a [[Measure space#null|null set]];
| |
| − | * it is [[Measure space#Completion|complete]], [[Measure space#Perfect and standard|perfect]], and the [[Hilbert space#L2 space|corresponding Hilbert space]] is separable.
| |
| | | | |
| − | ====References====
| + | import math; |
| | | | |
| − | {|
| + | int kmax=40; |
| − | |valign="top"|{{Ref|P}}|| Karl Petersen, "Ergodic theory", Cambridge (1983). {{MR|0833286}} {{ZBL|0507.28010}}
| + | |
| − | |-
| + | guide g; |
| − | |valign="top"|{{Ref|H1}}|| P.R. Halmos, "Measure theory", Van Nostrand (1950). {{MR|0033869}} {{ZBL|0040.16802}}
| + | for (int k=-kmax; k<=kmax; ++k) { |
| − | |-
| + | real phi = 0.2*k*pi; |
| − | |valign="top"|{{Ref|H2}}|| P.R. Halmos, "Lectures on ergodic theory", Math. Soc. Japan (1956). {{MR|0097489}} {{ZBL|0073.09302}}
| + | real rho = 1; |
| − | |-
| + | if (k!=0) { |
| − | |valign="top"|{{Ref|G}}|| Eli Glasner, "Ergodic theory via joinings", Amer. Math. Soc. (2003). {{MR|1958753}} {{ZBL|1038.37002}}
| + | rho = sin(phi)/phi; |
| − | |-
| + | } |
| − | |valign="top"|{{Ref|K}}|| Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995). {{MR|1321597}} {{ZBL|0819.04002}}
| + | pair z=rho*expi(phi); |
| − | |-
| + | g=g..z; |
| − | |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}}
| + | } |
| − | |-
| + | |
| − | |valign="top"|{{Ref|H}}|| Jean Haezendonck, "Abstract Lebesgue-Rohlin spaces", ''Bull. Soc. Math. de Belgique'' '''25''' (1973), 243–258. {{MR|0335733}} {{ZBL|0308.60006}}
| + | draw (g); |
| − | |-
| + | |
| − | |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}}
| + | defaultpen(0.75); |
| − | |-
| + | draw ( (0,0)--(1.3,0), dotted, Arrow(SimpleHead,5) ); |
| − | |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}}
| + | dot ( (1,0) ); |
| − | |-
| + | label ( "$a$", (1,0), NE ); |
| − | |valign="top"|{{Ref|F}}|| D.H. Fremlin, "Measure theory", Torres Fremlin, Colchester. Vol. 1: 2004 {{MR|2462519}} {{ZBL|1162.28001}}; Vol. 2: 2003 {{MR|2462280}} {{ZBL|1165.28001}}; Vol. 3: 2004 {{MR|2459668}} {{ZBL|1165.28002}}; Vol. 4: 2006 {{MR|2462372}} {{ZBL|1166.28001}}
| + | |
| − | |}
| + | </asy> |
