Hölder's inequality: Difference between revisions

Revision as of 06:00, 4 February 2014

In mathematics, the counting measure is an intuitive way to put a measure on any 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 ∞ if the subset is infinite.^[1]

The counting measure can be defined on any measurable set, but is mostly used on countable sets.^[1]

In formal notation, we can make any set $X$ into a measurable space by taking the sigma-algebra $\Sigma$ of measurable subsets to consist of all subsets of $X$ . Then the counting measure $\mu$ on this measurable space $(X,\Sigma )$ is the positive measure $\Sigma \rightarrow [0,+\infty ]$ defined by

\mu (A)={\begin{cases}\vert A\vert &{\text{if }}A{\text{ is finite}}\\+\infty &{\text{if }}A{\text{ is infinite}}\end{cases}}

for all $A\in \Sigma$ , where $\vert A\vert$ denotes the cardinality of the set $A$ .^[2]

The counting measure on $(X,\Sigma )$ is σ-finite if and only if the space $X$ is countable.^[3]

Notes

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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.

↑ ^1.0 ^1.1 Template:PlanetMath
↑ Schilling (2005), p.27
↑ Hansen (2009) p.47

[pm-1] 1.0 ^1.1 Template:PlanetMath

[2] Schilling (2005), p.27

[3] Hansen (2009) p.47

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[2]

[3]

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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>
+The counting measure can be defined on any measurable set, but is mostly used on [[countable]] sets.<ref name="pm" />
+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)]]

Hölder's inequality: Difference between revisions

Revision as of 06:00, 4 February 2014

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