Choquet capacity

A Choquet capacity, or just capacity, on a set $X$ is a kind of set function, mapping the power set $𝒫(X)$ to the real numbers.

Definition.

Let $\mathrm{F}$ be a collection of subsets of $X$. Then, an $\mathrm{F}$-capacity is an increasing set function

$$I:𝒫(X)\to {ℝ}_{+}$$

satisfying the following.

1. If ${(A_n)}_{n\in \mathbb{N}}$ is an increasing sequence of subsets of $X$ then $I(A_n)\to I\left({\bigcup }_m{A}_m\right)$ as $n\to \mathrm{\infty }$.

2. If ${(A_n)}_{n\in \mathbb{N}}$ is a decreasing sequence of subsets of $X$ such that $A_n\in ℱ$ for each $n$, then $I(A_n)\to I\left({\bigcap }_m{A}_m\right)$ as $n\to \mathrm{\infty }$.

The condition that $I$ is increasing means that $I(A)\le I(B)$ whenever $A\subseteq B$.Note that capacities differ from the concepts of measures and outer measures, as no additivity or subadditivity conditions are imposed. However, for any finite measure, there is a corresponding capacity (http://planetmath.org/CapacityGeneratedByAMeasure). An important application to the theory of measures and analytic sets is given by the capacitability theorem.

The $\mathrm{(}\mathrm{F}\mathrm{,}I\mathrm{)}$-capacitable sets are defined as follows. Recall that ${ℱ}_{\delta }$ denotes the collection of countable intersections of sets in the paving $ℱ$.

Definition.

Let $I$ be an $\mathrm{F}$-capacity on a set $X$. Then a subset $A\mathrm{\subseteq }X$ is $\mathrm{(}ℱ\mathrm{,}I\mathrm{)}$-capacitable if, for each $ϵ\mathrm{>}\mathrm{0}$, there exists a $B\mathrm{\in }{\mathrm{F}}_{\delta }$ such that $B\mathrm{\subseteq }A$ and $I\mathit{}\mathrm{(}B\mathrm{)}\mathrm{\ge }I\mathit{}\mathrm{(}A\mathrm{)}\mathrm{-}ϵ$.

Alternatively, such sets are called $I$-capacitable or, simply, capacitable.