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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク counting measure ： ウィキペディア英語版 counting measure In mathematics, the counting measure is an intuitive way to put a measure on any set: the "size" of a subset is taken to be the number of elements in the subset, if the subset has finitely many elements, and ∞ if the subset is infinite. The counting measure can be defined on any measurable set, but is mostly used on countable sets.〔 In formal notation, we can make any set ''X'' into a measurable space by taking the sigma-algebra $\backslash Sigma$ of measurable subsets to consist of all subsets of $X$. Then the counting measure $\backslash mu$ on this measurable space $(X,\backslash Sigma)$ is the positive measure $\backslash Sigma\backslash rightarrow()$ defined by :$$ \mu(A)=\begin \vert A \vert & \text A \text\\ +\infty & \text A \text \end for all $A\backslash in\backslash Sigma$, where $\backslash vert\; A\backslash vert$ denotes the cardinality of the set $A$.〔Schilling (2005), p.27〕 The counting measure on $(X,\backslash Sigma)$ is σ-finite if and only if the space $X$ is countable.〔Hansen (2009) p.47〕 ==Discussion== The counting measure is a special case of a more general construct. With the notation as above, any function $f\; \backslash colon\; X\; \backslash to\; [0,\; \backslash infty)$ defines a measure $\backslash mu$ on $(X,\; \backslash Sigma)$ via :$\backslash mu(A):=\backslash sum\_\; f(a)\backslash \; \backslash forall\; A\backslash subseteq\; X,$ where the possibly uncountable sum of real numbers is defined to be the sup of the sums over all finite subsets, i.e., :$\backslash sum\_\; y\; :=\; \backslash sup\_\; \backslash left\backslash .$ Taking ''f(x)=1'' for all ''x'' in ''X'' produces the counting measure. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「counting measure」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.