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

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====Criticism====
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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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A quote from {{Cite|Dur|Sect. 1.4(c), p. 33}}:
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====Notes====
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:
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<references />
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: $(S,\mathcal S)$ is said to be ''nice'' if there is a 1-1 map $\phi$ from $S$ into $\R$ so that $\phi$ and $\phi^{-1}$ are both measurable.
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:
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-------------------------------------------
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: Such spaces are often called ''standard Borel spaces,'' but we already have too many things named after Borel. The next result shows that most spaces arising in applications are nice.
 
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:
 
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: (4.12) ''Theorem.'' If $S$ is a Borel subset of a complete separable metric space $M$, and $\mathcal S$ is the collection of Borel subsets of $S$, then $(S,\mathcal S)$ is nice.
 
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It is not specified in the definition, whether $\phi(S)$ must be a Borel set, or not. The proof of the theorem provides just a Borel 1-1 map $\phi:S\to\R$ without addressing measurability of the function $\phi^{-1}$ and the set $\phi(S)$. (A complete proof would be considerably harder.) Later, in the proof of Theorem (1.6) of {{Cite|Dur|Sect. 4.1(c)}}, measurability of $\phi^{-1}$ and $\phi(S)$ is used (see the last line of the proof).
 
  
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====References====
 
  
 
{|
 
{|
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|valign="top"|{{Ref|Dur}}|| Richard Durrett, "Probability: theory and examples", second edition, Duxbury Press (1996).  &nbsp; {{MR|1609153}}
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| A || B || C
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|-
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| X || Y || Z
 
|}
 
|}
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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) );
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label ( "$a$", (1,0), NE );
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 +
</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


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

How to Cite This Entry:
Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=24265
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