capacity generated by a measure

Any finite measure (http://planetmath.org/SigmaFinite) can be extended to a set function on the power set of the underlying space. As the following result states, this will be a Choquet capacity.

Theorem.

Let $(X,\mathcal{F},\mu)$ be a finite measure space. Then,

	$\displaystyle\mu^{*}\colon\mathcal{P}(X)\to\mathbb{R}_{+},$
	$\displaystyle\mu^{*}(S)=\inf\left\{\mu(A)\colon A\in\mathcal{F},\ A\supseteq S\right\}$

is an $\mathcal{F}$ -capacity. Furthermore, a subset $S\subseteq X$ is $(\mathcal{F},\mu^{*})$ -capacitable if and only if it is in the completion (http://planetmath.org/CompleteMeasure) of $\mathcal{F}$ with respect to $\mu$ .

Note that, as well as being a capacity, $\mu^{*}$ is also an outer measure (see here (http://planetmath.org/ConstructionOfOuterMeasures)), which does not require the finiteness of $\mu$ . Clearly, $\mu^{*}(A)=\mu(A)$ for all $A\in\mathcal{F}$ , so $\mu^{*}$ is an extension of $\mu$ to the power set of $X$ , and is referred to as the outer measure generated by $\mu$ .

Recall that a subset $S\subseteq X$ is in the completion of $\mathcal{F}$ with respect to $\mu$ if and only if there are sets $A,B\in\mathcal{F}$ with $A\subseteq S\subseteq B$ and $\mu(B\setminus A)=0$ which, by the above theorem, is equivalent to the capacitability of $S$ .

Title	capacity generated by a measure
Canonical name	CapacityGeneratedByAMeasure
Date of creation	2013-03-22 18:47:35
Last modified on	2013-03-22 18:47:35
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	7
Author	gel (22282)
Entry type	Theorem
Classification	msc 28A05
Classification	msc 28A12
Synonym	outer measure generated by a measure
Defines	outer measure generated by