Namespaces
Variants
Actions

Difference between revisions of "Complete measure"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c0238001.png" /> on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c0238002.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c0238003.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c0238004.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c0238005.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c0238006.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c0238007.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c0238008.png" /> is the total variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c0238009.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380010.png" /> for a positive measure).
+
<!--
 +
c0238001.png
 +
$#A+1 = 32 n = 0
 +
$#C+1 = 32 : ~/encyclopedia/old_files/data/C023/C.0203800 Complete measure
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
 +
{{TEX|auto}}
 +
{{TEX|done}}
  
 +
A measure  $  \mu $
 +
on a  $  \sigma $-
 +
algebra  $  \Sigma $
 +
for which  $  A \in \Sigma $
 +
and  $  | \mu | ( A) = 0 $
 +
imply  $  E \in \Sigma $
 +
for every  $  E \subset  A $.
 +
Here  $  | \mu | $
 +
is the total variation of  $  \mu $(
 +
$  | \mu | = \mu $
 +
for a positive measure).
  
 
====Comments====
 
====Comments====
Complete measures arise as follows (cf. [[#References|[a1]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380011.png" /> be a set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380012.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380013.png" />-algebra of subsets of it and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380014.png" /> a positive measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380015.png" />. It may happen that some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380016.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380017.png" /> has a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380018.png" /> not belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380019.png" />. It is natural, then, to define the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380020.png" /> on such a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380021.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380022.png" />.
+
Complete measures arise as follows (cf. [[#References|[a1]]]). Let $  X $
 +
be a set, $  \Sigma $
 +
a $  \sigma $-
 +
algebra of subsets of it and $  \mu $
 +
a positive measure on $  \Sigma $.  
 +
It may happen that some set $  E \in \Sigma $
 +
with $  \mu ( E) = 0 $
 +
has a subset $  N $
 +
not belonging to $  \Sigma $.  
 +
It is natural, then, to define the measure $  \mu $
 +
on such a set $  N $
 +
as $  \mu ( N) = 0 $.
  
In general, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380023.png" /> be the collection of all sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380024.png" /> for which there exists sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380025.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380027.png" />. In this situation, define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380028.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380029.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380030.png" />-algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380031.png" /> becomes a complete measure on it (this process is called completion). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023800/c02380032.png" /> is then called a complete measure space.
+
In general, let $  \Sigma  ^ {*} $
 +
be the collection of all sets $  N $
 +
for which there exists sets $  E , F \in \Sigma $
 +
such that $  E \subset  N \subset  F $,  
 +
$  \mu ( F - E ) = 0 $.  
 +
In this situation, define $  \mu ( N) = 0 $.  
 +
Then $  \Sigma  ^ {*} $
 +
is a $  \sigma $-
 +
algebra and $  \mu $
 +
becomes a complete measure on it (this process is called completion). $  ( X , \Sigma  ^ {*} , \mu ) $
 +
is then called a complete measure space.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1974)  pp. 24</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" , Springer  (1965)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1974)  pp. 24</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" , Springer  (1965)</TD></TR></table>

Latest revision as of 17:45, 4 June 2020


A measure $ \mu $ on a $ \sigma $- algebra $ \Sigma $ for which $ A \in \Sigma $ and $ | \mu | ( A) = 0 $ imply $ E \in \Sigma $ for every $ E \subset A $. Here $ | \mu | $ is the total variation of $ \mu $( $ | \mu | = \mu $ for a positive measure).

Comments

Complete measures arise as follows (cf. [a1]). Let $ X $ be a set, $ \Sigma $ a $ \sigma $- algebra of subsets of it and $ \mu $ a positive measure on $ \Sigma $. It may happen that some set $ E \in \Sigma $ with $ \mu ( E) = 0 $ has a subset $ N $ not belonging to $ \Sigma $. It is natural, then, to define the measure $ \mu $ on such a set $ N $ as $ \mu ( N) = 0 $.

In general, let $ \Sigma ^ {*} $ be the collection of all sets $ N $ for which there exists sets $ E , F \in \Sigma $ such that $ E \subset N \subset F $, $ \mu ( F - E ) = 0 $. In this situation, define $ \mu ( N) = 0 $. Then $ \Sigma ^ {*} $ is a $ \sigma $- algebra and $ \mu $ becomes a complete measure on it (this process is called completion). $ ( X , \Sigma ^ {*} , \mu ) $ is then called a complete measure space.

References

[a1] W. Rudin, "Real and complex analysis" , McGraw-Hill (1974) pp. 24
[a2] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965)
How to Cite This Entry:
Complete measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_measure&oldid=46419
This article was adapted from an original article by A.P. Terekhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article