# extending a capacity to a Cartesian product

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

If $I$ is an $\mathrm{F}$-capacity and ${\pi}_{X}\mathrm{:}X\mathrm{\times}K\mathrm{\to}X$ is the projection map, we can form the composition^{}

$I\circ {\pi}_{X}:\mathcal{P}(X\times K)\to \mathbb{R},$ | ||

$I\circ {\pi}_{X}(S)=I({\pi}_{X}(S)).$ |

Then ${\pi}_{X}\mathit{}\mathrm{(}S\mathrm{)}\mathrm{\in}{\mathrm{F}}_{\delta}$ for any $S\mathrm{\in}{\mathrm{G}}_{\delta}$, and $I\mathrm{\circ}{\pi}_{X}$ is a $\mathrm{G}$-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 $\u03f5>0$, there is an $A\in {\mathcal{G}}_{\delta}$ such that $A\subseteq S$ and $I\circ {\pi}_{X}(A)>I\circ {\pi}_{X}(S)-\u03f5$. 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.

###### Lemma.

Let $S\mathrm{\subseteq}X\mathrm{\times}K$ be $\mathrm{(}\mathrm{G}\mathrm{,}I\mathrm{\circ}{\pi}_{X}\mathrm{)}$-capacitable. Then, ${\pi}_{X}\mathit{}\mathrm{(}S\mathrm{)}$ is $\mathrm{(}\mathrm{F}\mathrm{,}I\mathrm{)}$-capacitable.

Title | extending a capacity to a Cartesian product |
---|---|

Canonical name | ExtendingACapacityToACartesianProduct |

Date of creation | 2013-03-22 18:47:38 |

Last modified on | 2013-03-22 18:47:38 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 6 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 28A12 |

Classification | msc 28A05 |