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.

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.

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
Canonical name	ChoquetCapacity
Date of creation	2013-03-22 18:47:26
Last modified on	2013-03-22 18:47:26
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	5
Author	gel (22282)
Entry type	Definition
Classification	msc 28A12
Classification	msc 28A05
Synonym	capacity
Defines	capacitable