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Difference between revisions of "User:Boris Tsirelson/sandbox2"

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{|
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{{User:Ulf Rehmann/sandbox/Ref
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| Key = P
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| Author = Petersen, Karl
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| Booktitle = Ergodic theory
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| Publisher = Cambridge
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| Year  = 1983
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| Mrn = 0833286
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| Zbl = 0507.28010
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| tab = true
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}}
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|-
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|valign="top"|{{Ref|H1}}||  P.R. Halmos, "Measure theory", Van Nostrand (1950).    {{MR|0033869}}   {{ZBL|0040.16802}}
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|-
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|valign="top"|{{Ref|H2}}||  P.R. Halmos, "Lectures on ergodic theory", Math. Soc. Japan (1956).    {{MR|0097489}}   {{ZBL|0073.09302}}
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|-
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|valign="top"|{{Ref|G}}||  Eli Glasner, "Ergodic theory via joinings", Amer. Math. Soc. (2003).    {{MR|1958753}}   {{ZBL|1038.37002}}
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|-
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|valign="top"|{{Ref|K}}||  Alexander  S.  Kechris, "Classical      descriptive set theory",  Springer-Verlag  (1995).      {{MR|1321597}}    {{ZBL|0819.04002}}
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|-
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|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}}
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|-
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|valign="top"|{{Ref|S}}|| I.E. Segal,  "Abstract probability spaces and a theorem of Kolmogoroff",  ''Amer. J.  Math.'' '''76''' (1954), 721–732.   {{MR|0063602}}      {{ZBL|0056.12301}}
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|-
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|valign="top"|{{Ref|D}}|| L.E.  Dubins,  "Generalized random variables",  ''Trans. Amer. Math. Soc.''  '''84'''  (1957), 273–309.   {{MR|0085326}}      {{ZBL|0078.31003}}
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|-
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|valign="top"|{{Ref|W}}|| David  Williams, "Probability with martingales", Cambridge  (1991).      {{MR|1155402}}   {{ZBL|0722.60001}}
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|-
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|valign="top"|{{Ref|C}}||  Constantin Carathėodory, "Die homomorphieen von Somen und die  Multiplikation von  Inhaltsfunktionen" (German),  ''Annali della R.  Scuola Normale Superiore di Pisa (Ser. 2)'' '''8''' (1939), 105–130.      {{MR|1556820}}    {{ZBL|0021.11403}}
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|-
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|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}}
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|}
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*******************************************
  
 
[http://www.mediawiki.org/wiki/Extension_talk:MathJax#.5BWORKAROUND.5D_MathJax_works_in_preview.2C_but_not_in_page_display here]
 
[http://www.mediawiki.org/wiki/Extension_talk:MathJax#.5BWORKAROUND.5D_MathJax_works_in_preview.2C_but_not_in_page_display here]

Revision as of 18:25, 3 September 2013

[P] Petersen, Karl; Ergodic theory, (1983), Cambridge, 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
[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
[S] I.E. Segal, "Abstract probability spaces and a theorem of Kolmogoroff", Amer. J. Math. 76 (1954), 721–732.   MR0063602   Zbl 0056.12301
[D] L.E. Dubins, "Generalized random variables", Trans. Amer. Math. Soc. 84 (1957), 273–309.   MR0085326   Zbl 0078.31003
[W] David Williams, "Probability with martingales", Cambridge (1991).   MR1155402   Zbl 0722.60001
[C] Constantin Carathėodory, "Die homomorphieen von Somen und die Multiplikation von Inhaltsfunktionen" (German), Annali della R. Scuola Normale Superiore di Pisa (Ser. 2) 8 (1939), 105–130.   MR1556820   Zbl 0021.11403
[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


here

$$\text{ $K$ compact}$$

\[\text{ '"`UNIQ-MathJax2-QINU`"' compact}\]

\begin{equation} \mu (B)= \sup \{\mu(K): K\subset B, \text{ '"`UNIQ-MathJax3-QINU`"' compact}\}\, \end{equation}

and having the following property: \begin{equation}\label{e:tight} \mu (B)= \sup \{\mu(K): K\subset B, \mbox{ '"`UNIQ-MathJax4-QINU`"' compact}\}\, \end{equation} (see [Sc]).


The total variation measure of a $\mathbb C$-valued measure is defined on $\mathcal{B}$ as: \[ \abs{\mu}(B) :=\sup\left\{ \sum \abs{\mu(B_i)}: \text{$\{B_i\}\subset\mathcal{B}'"`UNIQ-MathJax7-QINU`"'B$}\right\}. \] In the real-valued case the above definition simplifies as


and the following identity holds: \begin{equation}\label{e:area_formula} \int_A J f (y) \, dy = \int_{\mathbb R^m} \mathcal{H}^0 (A\cap f^{-1} (\{z\}))\, d\mathcal{H}^n (z)\, . \end{equation}

Cp. with 3.2.2 of [EG]. From \eqref{e:area_formula} it is not difficult to conclude the following generalization (which also goes often under the same name):



\begin{equation}\label{ab} E=mc^2 \end{equation} By \eqref{ab}, it is possible. But see \eqref{ba} below: \begin{equation}\label{ba} E\ne mc^3, \end{equation} which is a pity.

How to Cite This Entry:
Boris Tsirelson/sandbox2. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox2&oldid=30028