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

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====Criticism====
 
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A quote from {{Cite|Dur|Sect. 1.4(c), p. 33}}:
 
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: $(S,\mathcal S)$ is said to be ''nice'' if there is a 1-1 map $\varphi$ from $S$ into $\R$ so that $\varphi$ and $\varphi^{-1}$ are both measurable.
 
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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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: (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 $\varphi(S)$ must be a Borel set, or not. The proof of the theorem provides just a Borel 1-1 map $\varphi:S\to\R$ without addressing measurability of the function $\varphi^{-1}$ and the set $\varphi(S)$. (A complete proof would be considerably harder.) Later, in the proof of Theorem (1.6) of {{Cite|Dur|Sect. 4.1(c)}}, measurability of $\varphi^{-1}$ and $\varphi(S)$ is used (see the last line of the proof).
 
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====References====
 
====References====
  
 
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|valign="top"|{{Ref|Dur}}|| Richard Durrett, "Probability: theory and examples", second edition, Duxbury Press (1996).    {{MR|1609153}}
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|valign="top"|{{Ref|Mac}}||   George W. Mackey, "Borel  structure in groups and their duals",   ''Trans.  Amer. Math. Soc.''  '''85''' (1957), 134–165.      {{MR|0089999}}      {{ZBL|0082.11201}}
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|valign="top"|{{Ref|Mal}}||Paul    Malliavin, "Integration and  probability", Springer-Verlag (1995).     {{MR|1335234}}   {{ZBL|0874.28001}}
 
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Revision as of 16:05, 17 April 2012

References

[Mac] George W. Mackey, "Borel structure in groups and their duals", Trans. Amer. Math. Soc. 85 (1957), 134–165.   MR0089999   Zbl 0082.11201
[Mal] Paul Malliavin, "Integration and probability", Springer-Verlag (1995).   MR1335234   Zbl 0874.28001
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Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=24267
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