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 $\mathrm{F}$ be a collection^{} of subsets of $X$. Then, an $\mathrm{F}$capacity is an increasing set function
$$I:\mathcal{P}(X)\to {\mathbb{R}}_{+}$$ 
satisfying the following.

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

2.
If ${({A}_{n})}_{n\in \mathbb{N}}$ is a decreasing sequence of subsets of $X$ such that ${A}_{n}\in \mathcal{F}$ 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 ${\mathcal{F}}_{\delta}$ denotes the collection of countable^{} intersections of sets in the paving $\mathcal{F}$.
Definition.
Let $I$ be an $\mathrm{F}$capacity on a set $X$. Then a subset $A\mathrm{\subseteq}X$ is $\mathrm{(}\mathcal{F}\mathrm{,}I\mathrm{)}$capacitable if, for each $\u03f5\mathrm{>}\mathrm{0}$, there exists a $B\mathrm{\in}{\mathrm{F}}_{\delta}$ such that $B\mathrm{\subseteq}A$ and $I\mathit{}\mathrm{(}B\mathrm{)}\mathrm{\ge}I\mathit{}\mathrm{(}A\mathrm{)}\mathrm{}\u03f5$.
Alternatively, such sets are called $I$capacitable or, simply, capacitable.
Title  Choquet capacity 

Canonical name  ChoquetCapacity 
Date of creation  20130322 18:47:26 
Last modified on  20130322 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 