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Difference between revisions of "User:Boris Tsirelson/sandbox1"

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$\newcommand{\Om}{\Omega}
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<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>
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\newcommand{\A}{\mathcal A}
 
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\newcommand{\B}{\mathcal B}
 
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\newcommand{\M}{\mathcal M} $
 
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The term '''"universally measurable"''' may be applied to
 
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* a [[measurable space]];
 
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* a subset of a measurable space;
 
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* a [[metric space]].
 
  
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Let $(X,\A)$ be a measurable space. A ''subset'' $A\subset X$ is called ''universally measurable'' if it is $\mu$-measurable for every finite measure $\mu$ on $(X,\A)$. In other words: $\mu_*(A)=\mu^*(A)$ where $\mu_*,\mu^*$ are the inner and outer measures for $\mu$, that is,
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====Notes====
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: $ \mu_*(A) = \max\{\mu(B):B\in\A,B\subset A\}\,,\quad
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<references />
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\mu^*(A) = \min\{\mu(B):B\in\A,B\supset A\}\,.$
 
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(See {{Cite|S|p. 170}}.)
 
  
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A separable ''metric space'' is called ''universally measurable'' if it is a universally measurable subset (as defined above) of its [[Metric space#completion|completion]]. Here the completion, endowed with the [[Measurable space#Borel sets|Borel σ-algebra]], is treated as a measurable space. (See {{Cite|S|p. 170}}, {{Cite|D|Sect. 11.5}}.)
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A ''measurable space''  is called ''universally measurable'' if it is [[Measurable space#isomorphic|isomorphic]] to some universally measurable metric space (as defined above) with the Borel σ-algebra. (See {{Cite|S|p. 171}}.)
 
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====References====
 
  
 
{|
 
{|
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|valign="top"|{{Ref|S}}|| Rae M. Shortt, "Universally measurable spaces: an invariance theorem and diverse characterizations", ''Fundamenta Mathematicae'' '''121''' (1984), 169–176.  &nbsp; {{MR|0765332}} &nbsp; {{ZBL|0573.28018}}
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| A || B || C
 
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|valign="top"|{{Ref|P}}|| David Pollard, "A user's guide to measure theoretic probability",  Cambridge (2002). &nbsp;  {{MR|1873379}} &nbsp;  {{ZBL|0992.60001}}
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| X || Y || Z
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|-
 
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|valign="top"|{{Ref|K}}|| Alexander  S.  Kechris, "Classical  descriptive set theory", Springer-Verlag  (1995). &nbsp;  {{MR|1321597}} &nbsp; {{ZBL|0819.04002}}
 
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|-
 
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|valign="top"|{{Ref|BK}}||  Howard Becker and Alexander S. Kechris, "The descriptive set theory of  Polish group actions", Cambridge (1996). &nbsp; {{MR|1425877}}  &nbsp;  {{ZBL|0949.54052}}
 
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|-
 
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|valign="top"|{{Ref|D}}||  Richard M. Dudley, "Real analysis and probability",  Wadsworth&Brooks/Cole (1989). &nbsp; {{MR|0982264}} &nbsp;  {{ZBL|0686.60001}}
 
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|-
 
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|valign="top"|{{Ref|M}}||  George  W.  Mackey,  "Borel structure in groups and their duals",  ''Trans.  Amer.  Math. Soc.''  '''85''' (1957), 134–165. &nbsp; {{MR|0089999}}  &nbsp; {{ZBL|0082.11201}}
 
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|-
 
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|valign="top"|{{Ref|H}}||  Paul R. Halmos, "Measure theory", v. Nostrand (1950). &nbsp;  {{MR|0033869}} &nbsp; {{ZBL|0040.16802}}
 
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|-
 
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|valign="top"|{{Ref|R}}||  Walter Rudin, "Principles of mathematical analysis", McGraw-Hill  (1953). &nbsp; {{MR|0055409}} &nbsp; {{ZBL|0052.05301}}
 
 
|}
 
|}
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-----------------------------------------
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-----------------------------------------
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$\newcommand*{\longhookrightarrow}{\lhook\joinrel\relbar\joinrel\rightarrow}$
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<asy>
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size(100,100);
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label(scale(1.7)*'$T(\\Sigma)\hookrightarrow T(\\Sigma,X)$',(0,0));
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</asy>
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<asy>
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size(220,220);
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import math;
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int kmax=40;
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guide g;
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for (int k=-kmax; k<=kmax; ++k) {
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  real phi = 0.2*k*pi;
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  real rho = 1;
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  if (k!=0) {
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    rho = sin(phi)/phi;
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  }
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  pair z=rho*expi(phi);
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  g=g..z;
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}
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draw (g);
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 +
defaultpen(0.75);
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draw ( (0,0)--(1.3,0), dotted, Arrow(SimpleHead,5) );
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dot ( (1,0) );
 +
label ( "$a$", (1,0), NE );
 +
 +
</asy>

Latest revision as of 07:12, 13 March 2016

[1]

Notes

  1. ↑ http://hea-www.harvard.edu/AstroStat; http://www.incagroup.org ; http://astrostatistics.psu.edu


A B C
X Y Z


$\newcommand*{\longhookrightarrow}{\lhook\joinrel\relbar\joinrel\rightarrow}$

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Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=21111
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