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# 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. 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.

## Mathematics Subject Classification

28A12*no label found*28A05

*no label found*

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