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Choquet capacity
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(Definition)
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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 \begin{equation*} I\colon\mathcal{P}(X)\rightarrow\mathbb{R}_+ \end{equation*}satisfying the following.
- 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 $n\rightarrow\infty$
- 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 $n\rightarrow\infty$
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. 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 $I(B)\ge I(A)-\epsilon$
Alternatively, such sets are called $I$ capacitable or, simply, capacitable.
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"Choquet capacity" is owned by gel.
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| Also defines: |
capacitable |
| Keywords: |
paved space, set function |
This object's parent.
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Cross-references: paving, intersections of sets, countable, capacitability theorem, analytic sets, additivity, outer measures, measures, real numbers, power set
There are 21 references to this entry.
This is version 2 of Choquet capacity, born on 2009-02-01, modified 2009-02-01.
Object id is 11587, canonical name is ChoquetCapacity.
Accessed 1165 times total.
Classification:
| AMS MSC: | 28A05 (Measure and integration :: Classical measure theory :: Classes of sets , measurable sets, Suslin sets, analytic sets) | | | 28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities) |
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Pending Errata and Addenda
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