# Choquet capacity

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

###### Definition.

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

 $I\colon\mathcal{P}(X)\rightarrow\mathbb{R}_{+}$

satisfying the following.

1. 1.

If $(A_{n})_{n\in\mathbb{N}}$ is an increasing sequence of subsets of $X$ then $I(A_{n})\rightarrow I\left(\bigcup_{m}A_{m}\right)$ as $n\rightarrow\infty$.

2. 2.

If $(A_{n})_{n\in\mathbb{N}}$ is a decreasing sequence of subsets of $X$ such that $A_{n}\in\mathcal{F}$ for each $n$, then $I(A_{n})\rightarrow I\left(\bigcap_{m}A_{m}\right)$ as $n\rightarrow\infty$.

The condition that $I$ is increasing means that $I(A)\leq I(B)$ 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 $(\mathcal{F},I)$-capacitable sets are defined as follows. Recall that $\mathcal{F}_{\delta}$ denotes the collection of countable intersections of sets in the paving $\mathcal{F}$.

###### Definition.

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

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

Title Choquet capacity ChoquetCapacity 2013-03-22 18:47:26 2013-03-22 18:47:26 gel (22282) gel (22282) 5 gel (22282) Definition msc 28A12 msc 28A05 capacity capacitable