Any finite measure 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,

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 of $\mathcal{F}$ with respect to $\mu$ .

Note that, as well as being a capacity, $\mu^*$ is also an outer measure (see here), 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 $S$ .