# 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 CapacityGeneratedByAMeasure 2013-03-22 18:47:35 2013-03-22 18:47:35 gel (22282) gel (22282) 7 gel (22282) Theorem msc 28A05 msc 28A12 outer measure generated by a measure outer measure generated by