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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 Signed measure ： ウィキペディア英語版 Signed measure In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values. Some authors may call it a charge,〔A charge need not be countably additive: it can only be finitely additive. See reference for a comprehensive introduction to the subject.〕 by analogy with electric charge, which is a familiar distribution that takes on positive and negative values. ==Definition== There are two slightly different concepts of a signed measure, depending on whether or not one allows it to take infinite values. In research papers and advanced books signed measures are usually only allowed to take finite values, while undergraduate textbooks often allow them to take infinite values. To avoid confusion, this article will call these two cases "finite signed measures" and "extended signed measures". Given a measurable space (''X'', Σ), that is, a set ''X'' with a sigma algebra Σ on it, an extended signed measure is a function :$\backslash mu:\backslash Sigma\backslash to\; \backslash mathbb\; \backslash cup\backslash$ such that $\backslash mu(\backslash emptyset)=0$ and $\backslash mu$ is sigma additive, that is, it satisfies the equality :$\backslash mu\backslash left(\backslash bigcup\_^\backslash infty\; A\_n\backslash right)\; =\; \backslash sum\_^\backslash infty\; \backslash mu(A\_n)$ for any sequence ''A''1, ''A''2, ..., ''A''''n'', ... of disjoint sets in Σ. One consequence is that any extended signed measure can take +∞ as value, or it can take −∞ as value, but both are not available. The expression ∞ − ∞ is undefined 〔See the article "''Extended real number line''" for more information.〕 and must be avoided. A finite signed measure is defined in the same way, except that it is only allowed to take real values. That is, it cannot take +∞ or −∞. Finite signed measures form a vector space, while extended signed measures are not even closed under addition, which makes them rather hard to work with. On the other hand, measures are extended signed measures, but are not in general finite signed measures. 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「Signed 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.