Choquet capacity
ChoquetCapacity
2009-02-01 04:12:45
2009-02-01 19:37:53
Definition
analytic set
capacitable
paved space
set function
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A \emph{Choquet capacity}, or just \emph{capacity}, on a set $X$ is a kind of set function, mapping the power set $\mathcal{P}(X)$ to the real numbers.
\begin{definition*}
Let $\mathcal{F}$ be a collection of subsets of $X$. Then, an $\mathcal{F}$-capacity is an increasing set function
\begin{equation*}
I\colon\mathcal{P}(X)\rightarrow\mathbb{R}_+
\end{equation*}
satisfying the following.
\begin{enumerate}
\item If $(A_n)_{n\in\mathbb{N}}$ is an increasing sequence of subsets of $X$ then $I(A_n)\rightarrow I\left(\bigcup_mA_m\right)$ as $n\rightarrow\infty$.
\item 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 $I(A_n)\rightarrow I\left(\bigcap_mA_m\right)$ as $n\rightarrow\infty$.
\end{enumerate}
\end{definition*}
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 \PMlinkname{corresponding capacity}{CapacityGeneratedByAMeasure}. An important application to the theory of measures and analytic sets is given by the capacitability theorem.
The \emph{$(\mathcal{F},I)$-capacitable} sets are defined as follows. Recall that $\mathcal{F}_\delta$ denotes the collection of countable intersections of sets in the paving $\mathcal{F}$.
\begin{definition*}
Let $I$ be an $\mathcal{F}$-capacity on a set $X$. Then a subset $A\subseteq X$ is \emph{$(\mathcal{F},I)$-capacitable} if, for each $\epsilon >0$, there exists a $B\in\mathcal{F}_\delta$ such that $B\subseteq A$ and $I(B)\ge I(A)-\epsilon$.
\end{definition*}
Alternatively, such sets are called $I$-capacitable or, simply, capacitable.