Hölder's inequality: Difference between revisions

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In [[mathematics]], the '''counting measure''' is an intuitive way to put a [[measure (mathematics)|measure]] on any [[Set (mathematics)|set]]: the "size" of a [[subset]] is taken to be the number of elements in the subset, if the subset has finitely many elements, and [[&infin;]] if the subset is [[infinite_set|infinite]].<ref name="pm">{{PlanetMath|urlname=CountingMeasure|title=Counting Measure}}</ref>
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The counting measure can be defined on any measurable set, but is mostly used on [[countable]] sets.<ref name="pm" />
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In formal notation, we can make any set <math>X</math> into a [[measurable space]] by taking the [[sigma-algebra]] <math>\Sigma </math> of measurable subsets to consist of all subsets of <math>X</math>.   Then the counting measure <math>\mu</math> on this measurable space <math>(X,\Sigma)</math> is the positive measure <math>\Sigma\rightarrow[0,+\infty]</math> defined by
 
:<math>
\mu(A)=\begin{cases}
\vert A \vert & \text{if } A \text{ is finite}\\
+\infty & \text{if } A \text{ is infinite}
\end{cases}
</math>
 
for all <math>A\in\Sigma</math>, where <math>\vert A\vert</math> denotes the [[cardinality]] of the set <math>A</math>.<ref>Schilling (2005), p.27</ref>
 
The counting measure on <math>(X,\Sigma)</math> is [[σ-finite]] if and only if the space <math>X</math> is [[countable]].<ref>Hansen (2009) p.47</ref>
 
==Notes==
{{Reflist}}
==References==
*Schilling, René L. (2005)."Measures, Integral and Martingales". Cambridge University Press.
*Hansen, Ernst (2009)."Measure theory, Fourth Edition". Department of Mathematical Science, University of Copenhagen.
{{DEFAULTSORT:Counting Measure}}
[[Category:Measures (measure theory)]]

Revision as of 10:02, 1 March 2014

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