# proof of Choquet’s capacitability theorem

Let $(X,\mathcal{F})$ be a paved space such that $\mathcal{F}$ is closed under finite unions and finite intersections^{}, and let $I$ be an $\mathcal{F}$-capacity (http://planetmath.org/ChoquetCapacity). We prove the capacitability theorem, which states that all $\mathcal{F}$-analytic (http://planetmath.org/AnalyticSet2) sets are $(\mathcal{F},I)$-capacitable (http://planetmath.org/ChoquetCapacity). The idea is to deduce it from the following special case.

###### Lemma.

With the above notation, every set in ${\mathrm{F}}_{\sigma \mathit{}\delta}$ is $\mathrm{(}\mathrm{F}\mathrm{,}I\mathrm{)}$-capacitable.

Recall that ${\mathcal{F}}_{\sigma \delta}$ is the collection^{} of countable^{} intersections of countable unions in $\mathcal{F}$ and, since countable unions and intersections of analytic sets are analytic, all such sets are analytic. According to the capacitability theorem they should then be capacitable, and the lemma is indeed a special case.

Supposing that the lemma is true, then the capacitability theorem can be deduced as follows. For an $\mathcal{F}$-analytic set^{} $A\subseteq X$, there is a compact paved space (http://planetmath.org/PavedSpace) $(K,\mathcal{K})$ and $S\in {(\mathcal{F}\times \mathcal{K})}_{\sigma \delta}$ such that $A={\pi}_{X}(S)$, where ${\pi}_{X}$ is the projection map from $X\times K$ to $X$. Letting $\mathcal{G}$ be the closure under finite unions and finite intersections of the paving $\mathcal{F}\times \mathcal{K}$, then the composition^{} $I\circ {\pi}_{X}$ is a $\mathcal{G}$-capacity, and the projection of any $(\mathcal{G},I\circ {\pi}_{X})$-capacitable set onto $X$ is itself $(\mathcal{F},I)$-capacitable (see extending a capacity to a Cartesian product (http://planetmath.org/ExtendingACapacityToACartesianProduct)). In particular, $S\in {\mathcal{G}}_{\sigma \delta}$ so, by the lemma, is $(\mathcal{G},I\circ {\pi}_{X})$-capacitable. Therefore, $A={\pi}_{X}(S)$ is $(\mathcal{F},I)$-capacitable. It only remains to prove the lemma.

###### Proof of lemma.

If $S\in {\mathcal{F}}_{\sigma \delta}$ then there exists ${S}_{m,n}\in \mathcal{F}$ such that

$$S=\bigcap _{n}\bigcup _{m}{S}_{m,n}.$$ |

For any positive integers ${m}_{1},{m}_{2},\mathrm{\dots},{m}_{k}$ let us write

$$S({m}_{1},{m}_{2},\mathrm{\dots},{m}_{k})\equiv \left(\bigcap _{n\le k}\bigcup _{m\le {m}_{n}}{S}_{m,n}\right)\bigcap \left(\bigcap _{n>k}\bigcup _{m}{S}_{m,n}\right).$$ |

In particular, $S()=S$ and, $I(S())=I(S)$. For any $\u03f5>0$ and $k\in \mathbb{N}$ suppose that we have chosen positive integers ${m}_{1},\mathrm{\dots},{m}_{k-1}$ such that $I(S({m}_{1},\mathrm{\dots},{m}_{k-1}))>I(S)-\u03f5$. Since $I$ is a capacity and $S({m}_{1},\mathrm{\dots},{m}_{k})$ increases to $S({m}_{1},\mathrm{\dots},{m}_{k-1})$ as ${m}_{k}$ increases to infinity^{},

$$I(S({m}_{1},\mathrm{\dots},{m}_{k}))\to I(S({m}_{1},\mathrm{\dots},{m}_{k-1}))$$ |

as ${m}_{k}$ tends to infinity. So, by choosing ${m}_{k}$ large enough, we have

$$I(S({m}_{1},\mathrm{\dots},{m}_{k}))>I(S)-\u03f5.$$ |

Then, by induction^{}, we can find an infinite^{} sequence ${m}_{1},{m}_{2},\mathrm{\dots}$ such that this inequality holds for every $k$.
Setting

${A}_{k}\equiv {\displaystyle \bigcap _{n\le k}}{\displaystyle \bigcup _{m\le {m}_{n}}}{S}_{m,n}\in \mathcal{F},$ | ||

$A\equiv {\displaystyle \bigcap _{n}}{\displaystyle \bigcup _{m\le {m}_{n}}}{S}_{m,n}={\displaystyle \bigcap _{k}}{A}_{k}\in {\mathcal{F}}_{\delta},$ |

then $A\subseteq S$. Furthermore, ${A}_{k}$ contains $S({m}_{1},\mathrm{\dots},{m}_{k})$ and decreases to $A$ as $k$ tends to infinity. As $I$ is an $\mathcal{F}$-capacity this gives

$$I(A)=\underset{k\to \mathrm{\infty}}{lim}I({A}_{k})\ge \underset{k\to \mathrm{\infty}}{lim}I(S({m}_{1},\mathrm{\dots},{m}_{k}))\ge I(S)-\u03f5.$$ |

So $S$ is $(\mathcal{F},I)$-capacitable, as required. ∎

Title | proof of Choquet’s capacitability theorem |
---|---|

Canonical name | ProofOfChoquetsCapacitabilityTheorem |

Date of creation | 2013-03-22 18:47:51 |

Last modified on | 2013-03-22 18:47:51 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 4 |

Author | gel (22282) |

Entry type | Proof |

Classification | msc 28A05 |

Classification | msc 28A12 |