A capacity on a set $X$ can be extended to a set function on a Cartesian product $X\times K$ simply by projecting any subset onto $X$ , and then applying the original capacity.

Theorem   Suppose that $(X,\mathcal{F})$ is a paved space such that $\mathcal{F}$ is closed under finite unions and finite intersections, and that $(K,\mathcal{K})$ is a compact paved space. Define $\mathcal{G}$ to be the closure under finite unions and finite intersections of the paving $\mathcal{F}\times\mathcal{K}$ on $X\times K$ .

If $I$ is an $\mathcal{F}$ -capacity and $\pi_X\colon X\times K\to X$ is the projection map, we can form the composition

Then $\pi_X(S)\in \mathcal{F}_\delta$ for any $S\in\mathcal{G}_\delta$ , and $I\circ\pi_X$ is a $\mathcal{G}$ -capacity.

This result justifies looking at capacities when considering projections from the Cartesian product $X\times K$ onto $X$ . We see that the property of being a capacity is preserved under composing with such projections. However, additivity of set functions is not preserved, so the corresponding result would not be true if ``capacity'' was replaced by ``measure'' or ``outer measure''.

Recall that if $S\subseteq X\times K$ is $(\mathcal{G},I\circ\pi_X)$ -capacitable then, for any $\epsilon>0$ , there is an $A\in\mathcal{G}_\delta$ such that $A\subseteq S$ and $I\circ\pi_X(A)>I\circ\pi_X(S)-\epsilon$ . However, $\pi_X(A)\subseteq\pi_X(S)$ and, by the above theorem, $\pi_X(A)\in\mathcal{F}_\delta$ . This has the following consequence.