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In mathematics — specifically, in geometric measure theory — spherical measure ''σ''''n'' is the “natural” Borel measure on the ''n''-sphere S''n''. Spherical measure is often normalized so that it is a probability measure on the sphere, i.e. so that ''σ''''n''(S''n'') = 1. ==Definition of spherical measure== There are several ways to define spherical measure. One way is to use the usual “round” or “arclength” metric ''ρ''''n'' on S''n''; that is, for points ''x'' and ''y'' in S''n'', ''ρ''''n''(''x'', ''y'') is defined to be the (Euclidean) angle that they subtend at the centre of the sphere (the origin of R''n''+1). Now construct ''n''-dimensional Hausdorff measure ''H''''n'' on the metric space (S''n'', ''ρ''''n'') and define : One could also have given S''n'' the metric that it inherits as a subspace of the Euclidean space R''n''+1; the same spherical measure results from this choice of metric. Another method uses Lebesgue measure ''λ''''n''+1 on the ambient Euclidean space R''n''+1: for any measurable subset ''A'' of S''n'', define ''σ''''n''(''A'') to be the (''n'' + 1)-dimensional volume of the “wedge” in the ball B''n''+1 that it subtends at the origin. That is, : where : The fact that all these methods define the same measure on S''n'' follows from an elegant result of Christensen: all these measures are obviously uniformly distributed on S''n'', and any two uniformly distributed Borel regular measures on a separable metric space must be constant (positive) multiples of one another. Since all our candidate ''σ''''n''’s have been normalized to be probability measures, they are all the same 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「Spherical measure」の詳細全文を読む スポンサード リンク
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