# proof of extending a capacity to a Cartesian product

Let $(X,\mathcal{F})$ be a paved space such that $\mathcal{F}$ is closed under^{} finite unions and finite intersections^{}, and $(K,\mathcal{K})$ be 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 $X\times K$.
For an $\mathcal{F}$-capacity $I$, define

$\stackrel{~}{I}:\mathcal{P}(X\times K)\to \mathbb{R},$ | ||

$\stackrel{~}{I}(S)=I({\pi}_{X}(S)),$ |

where ${\pi}_{X}$ is the projection map onto $X$. We show that $\stackrel{~}{I}$ is a $\mathcal{G}$-capacity and that ${\pi}_{X}(S)\in {\mathcal{F}}_{\delta}$ whenever $S\in {\mathcal{G}}_{\delta}$.

Clearly, the property that $\stackrel{~}{I}$ is an increasing set function follows from the fact that $I$ satisfies this property. Furthermore, if ${S}_{n}\subseteq X\times K$ is an increasing sequence of sets with $S={\bigcup}_{n}{S}_{n}$ then ${\pi}_{X}({S}_{n})$ is an increasing sequence and

$$\stackrel{~}{I}(S)=I({\pi}_{X}(S))=I\left(\bigcup _{n}{\pi}_{X}({S}_{n})\right)=\underset{n\to \mathrm{\infty}}{lim}I({\pi}_{X}({S}_{n}))=\underset{n\to \mathrm{\infty}}{lim}\stackrel{~}{I}({S}_{n}).$$ |

To prove that $\stackrel{~}{I}$ is a $\mathcal{G}$-capacity, it only remains to show that if ${S}_{n}$ is a sequence^{} in $\mathcal{G}$ decreasing to $S\subseteq X\times K$ then $\stackrel{~}{I}({S}_{n})\to \stackrel{~}{I}(S)$.
Note that any $S$ in $\mathcal{G}$ can be written as $S={\bigcap}_{j=1}^{m}{\bigcup}_{k=1}^{{n}_{j}}{A}_{j,k}\times {K}_{j,k}$ for sets ${A}_{j,k}\in \mathcal{F}$ and ${K}_{j,k}\in \mathcal{K}$. The projection onto $X$ is then

$${\pi}_{X}(S)=\bigcup \{\bigcap _{j=1}^{m}{A}_{j,{k}_{j}}:{k}_{j}\le {n}_{j},\bigcap _{j=1}^{m}{K}_{j,{k}_{j}}\ne \mathrm{\varnothing}\}$$ |

which, as $\mathcal{F}$ is closed under finite unions and finite intersections, must be in $\mathcal{F}$. Furthermore, for any $x\in X$,

$${S}_{x}\equiv \{y\in K:(x,y)\in S\}=\bigcap _{j=1}^{m}\bigcup \{{K}_{j,k}:k\le {n}_{j},x\in {A}_{j,k}\}.$$ |

This shows that ${S}_{x}$ is in the closure ${\mathcal{K}}^{*}$ of $\mathcal{K}$ under finite unions and finite intersections. Furthermore, since compact pavings are closed subsets of a compact topological space^{} (http://planetmath.org/CompactPavingsAreClosedSubsetsOfACompactSpace), ${\mathcal{K}}^{*}$ is itself a compact paving.

Now let ${S}_{n}$ be a decreasing sequence of sets in $\mathcal{G}$ and set $S={\bigcap}_{n}{S}_{n}$. Then ${\pi}_{X}(S)\subseteq {\pi}_{X}({S}_{n})$ for each $n$, giving ${\pi}_{X}(S)\subseteq {\bigcap}_{n}{\pi}_{X}({S}_{n})$. To prove the reverse inequality, consider $x\in {\bigcap}_{n}{\pi}_{X}({S}_{n})$. Then, ${({S}_{n})}_{x}$ is a nonempty set in ${\mathcal{K}}^{*}$ for all $n$. By compactness, ${S}_{x}={\bigcap}_{n}{({S}_{n})}_{x}$ must also be nonempty and therefore $x\in {\pi}_{X}(S)$. This shows that

$$\bigcap _{n}{\pi}_{X}({S}_{n})={\pi}_{X}(S).$$ |

Furthermore, as we have shown that ${\pi}_{X}({S}_{n})\in \mathcal{F}$ and, as $I$ is an $\mathcal{F}$-capacity,

$$\stackrel{~}{I}({S}_{n})=I({\pi}_{X}({S}_{n}))\to I({\pi}_{X}(S))=\stackrel{~}{I}(S).$$ |

So $\stackrel{~}{I}$ is a $\mathcal{G}$-capacity.

We finally show that if $S\in {\mathcal{G}}_{\delta}$ then ${\pi}_{X}(S)\in {\mathcal{F}}_{\delta}$. By definition, there is a sequence ${S}_{n}\in \mathcal{G}$ such that $S={\bigcap}_{n}{S}_{n}$. Setting ${S}_{n}^{\prime}={\bigcap}_{m\le n}{S}_{m}$ then, since $\mathcal{G}$ is closed under finite unions and finite intersections, ${S}_{n}^{\prime}\in \mathcal{G}$. Furthermore, ${S}_{n}^{\prime}$ decreases to $S$ so, as shown above, ${\pi}_{X}({S}_{n}^{\prime})\in \mathcal{F}$ and

$${\pi}_{X}(S)=\bigcap _{n}{\pi}_{X}({S}_{n}^{\prime})\in {\mathcal{F}}_{\delta}$$ |

as required.

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

Canonical name | ProofOfExtendingACapacityToACartesianProduct |

Date of creation | 2013-03-22 18:47:41 |

Last modified on | 2013-03-22 18:47:41 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 5 |

Author | gel (22282) |

Entry type | Proof |

Classification | msc 28A12 |

Classification | msc 28A05 |