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

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====On terminology====
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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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The term "standard probability space" is used in {{Cite|I}}. The same, or very similar, notion appears also as: "Lebesgue space" {{Cite|Ro}}, {{Cite|Ru}}, {{Cite|P}}, {{Cite|G}}; "standard Lebesgue space" {{Cite|G}}; "Lebesgue-Rohlin space" {{Cite|H}}, {{Cite|B}}; and "L. R. space" {{Cite|H}}.
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====Notes====
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<references />
  
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Some authors admit totally finite (not necessarily probability) measures {{Cite|P}},  {{Cite|B}}. Note also "standard σ-finite measure" in {{Cite|M}}. Some authors exclude spaces of cardinality higher than continuum ({{Cite|Ro}}, {{Cite|Ru}}, {{Cite|G}}, but not {{Cite|I}}, {{Cite|H}}, {{Cite|M}},  {{Cite|P}}, {{Cite|B}}) even if it is isomorphic mod 0 to $(0,1)$ with Lebesgue measure.
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====References====
 
  
 
{|
 
{|
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|valign="top"|{{Ref|I}}||   Kiyosi Itô, "Introduction to  probability theory", Cambridge (1984).  &nbsp; {{MR|0777504}}  &nbsp; {{ZBL|0545.60001}}
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| A || B || C
 
|-
 
|-
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|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. &nbsp;  {{MR|1308547}} &nbsp;  {{ZBL|0788.60001}}
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| X || Y || Z
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|-
 
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|valign="top"|{{Ref|H}}||  Jean  Haezendonck, "Abstract  Lebesgue-Rohlin  spaces",  ''Bull. Soc.  Math.  de Belgique'' '''25'''  (1973), 243–258.  &nbsp;    {{MR|0335733}}  &nbsp;  {{ZBL|0308.60006}}
 
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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.    &nbsp;  {{MR|0006617}} &nbsp;    {{ZBL|0063.01888}}
 
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|-
 
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|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. &nbsp; {{MR|0047744}} &nbsp; Translated from Russian:  Рохлин, В. А. (1949), "Об основных понятиях теории меры", Математический  Сборник (Новая Серия) 25(67): 107–150. &nbsp; {{MR|0030584}}
 
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|-
 
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|valign="top"|{{Ref|P}}||  Karl Petersen, "Ergodic theory", Cambridge  (1983). &nbsp; {{MR|0833286}} &nbsp; {{ZBL|0507.28010}}
 
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|-
 
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|valign="top"|{{Ref|G}}|| Eli Glasner,  "Ergodic theory via joinings", Amer. Math. Soc. (2003). &nbsp;  {{MR|1958753}} &nbsp; {{ZBL|1038.37002}}
 
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|-
 
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|valign="top"|{{Ref|B}}||  V.I. Bogachev, "Measure theory",    Springer-Verlag (2007). &nbsp;  {{MR|2267655}}    &nbsp;{{ZBL|1120.28001}}
 
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|-
 
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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. &nbsp;  {{MR|0089999}} &nbsp;    {{ZBL|0082.11201}}
 
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|-
 
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|valign="top"|{{Ref|Mal}}||Paul  Malliavin, "Integration and  probability", Springer-Verlag (1995).  &nbsp; {{MR|1335234}}  &nbsp;  {{ZBL|0874.28001}}
 
 
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|}
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-----------------------------------------
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-----------------------------------------
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 +
$\newcommand*{\longhookrightarrow}{\lhook\joinrel\relbar\joinrel\rightarrow}$
 +
 +
<asy>
 +
size(100,100);
 +
label(scale(1.7)*'$T(\\Sigma)\hookrightarrow T(\\Sigma,X)$',(0,0));
 +
</asy>
 +
 +
<asy>
 +
size(220,220);
 +
 +
import math;
 +
 +
int kmax=40;
 +
 +
guide g;
 +
for (int k=-kmax; k<=kmax; ++k) {
 +
  real phi = 0.2*k*pi;
 +
  real rho = 1;
 +
  if (k!=0) {
 +
    rho = sin(phi)/phi;
 +
  }
 +
  pair z=rho*expi(phi);
 +
  g=g..z;
 +
}
 +
 
 +
draw (g);
 +
 +
defaultpen(0.75);
 +
draw ( (0,0)--(1.3,0), dotted, Arrow(SimpleHead,5) );
 +
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=24227
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