Namespaces
Variants
Actions

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

From Encyclopedia of Mathematics
Jump to: navigation, search
 
(159 intermediate revisions by the same user not shown)
Line 1: Line 1:
−
'''Measure algebra''' may refer to:
+
<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>
  
−
* algebra of measures on a topological group with the operation of convolution; see [[measure algebra (harmonic analysis)]];
+
====Notes====
−
* normed Boolean algebra, either in general or consisting of equivalence classes of measurable sets; see [[measure algebra (measure theory)]].
+
<references />
  
−
=Measure algebra (measure theory)=
+
-------------------------------------------
  
−
{{MSC.|28Axx|28A50,60A10}}
 
  
−
[[:Category:Classical measure theory]]
+
{|
 +
| A || B || C
 +
|-
 +
| X || Y || Z
 +
|}
 +
 
 +
 
 +
 
 +
-----------------------------------------
 +
-----------------------------------------
 +
 
 +
$\newcommand*{\longhookrightarrow}{\lhook\joinrel\relbar\joinrel\rightarrow}$
 +
 
 +
<asy>
 +
size(100,100);
 +
label(scale(1.7)*'$T(\\Sigma)\hookrightarrow T(\\Sigma,X)$',(0,0));
 +
</asy>
  
−
{{TEX|done}}
+
<asy>
 +
size(220,220);
  
−
$\newcommand{\Om}{\Omega}
+
import math;
−
\newcommand{\om}{\omega}
 
−
\newcommand{\F}{\mathcal F}
 
−
\newcommand{\B}{\mathcal B}
 
−
\newcommand{\M}{\mathcal M} $
 
−
A '''measure algebra''' is a pair $(B,\mu)$ where $B$ is a Boolean σ-algebra and $\mu$ is a (strictly) positive measure on $B$. The (strict) positivity means $\mu(x)\ge0$ and $\mu(x)=0\iff x=\bszero_B$ for all $x\in B$. However, about the greatest value $\mu(\bsone_B)$ of $\mu$, assumptions differ from $\mu(\bsone_B)=1$ (that is, $\mu$ is a probability measure) in {{Cite|Ha2|p. 43}} and {{Cite|K|Sect. 17.F}} to $\mu(\bsone_B)<\infty$ (that is, $\mu$ is a totally finite measure) in {{Cite|G|Sect. 2.1}} to $\mu(\bsone_B)\le\infty$ in {{Cite|P|Sect. 1.4C}} and {{Cite|Ha1|Sect. 40}}.
 
  
 +
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);
  
−
====References====
+
defaultpen(0.75);
 +
draw ( (0,0)--(1.3,0), dotted, Arrow(SimpleHead,5) );
 +
dot ( (1,0) );
 +
label ( "$a$", (1,0), NE );
  
−
{|
+
</asy>
−
|valign="top"|{{Ref|P}}||  Karl Petersen, "Ergodic theory", Cambridge (1983). &nbsp; {{MR|0833286}} &nbsp; {{ZBL|0507.28010}}
 
−
|-
 
−
|valign="top"|{{Ref|H1}}|| P.R. Halmos, "Measure theory", Van Nostrand (1950). &nbsp; {{MR|0033869}} &nbsp; {{ZBL|0040.16802}}
 
−
|-
 
−
|valign="top"|{{Ref|H2}}|| P.R. Halmos, "Lectures on ergodic theory", Math. Soc. Japan (1956). &nbsp; {{MR|0097489}} &nbsp; {{ZBL|0073.09302}}
 
−
|-
 
−
|valign="top"|{{Ref|G}}|| Eli Glasner, "Ergodic theory via joinings", Amer. Math. Soc. (2003). &nbsp; {{MR|1958753}} &nbsp; {{ZBL|1038.37002}}
 
−
|-
 
−
|valign="top"|{{Ref|K}}|| Alexander  S.  Kechris, "Classical    descriptive set theory", Springer-Verlag  (1995). &nbsp;  {{MR|1321597}} &nbsp; {{ZBL|0819.04002}}
 
−
|-
 
−
|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}}
 
−
|-
 
−
|valign="top"|{{Ref|F}}||  D.H. Fremlin, "Measure theory", Torres  Fremlin, Colchester. Vol. 1:  2004 &nbsp; {{MR|2462519}} &nbsp;  {{ZBL|1162.28001}}; Vol. 2:  2003  &nbsp; {{MR|2462280}}  &nbsp; {{ZBL|1165.28001}}; Vol. 3:  2004 &nbsp;  {{MR|2459668}}  &nbsp; {{ZBL|1165.28002}}; Vol. 4:  2006 &nbsp;    {{MR|2462372}} &nbsp; {{ZBL|1166.28001}}
 
−
|}
 

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=21688
Retrieved from "https://encyclopediaofmath.org/index.php?title=User:Boris_Tsirelson/sandbox1&oldid=37839"