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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 Tangent measure ： ウィキペディア英語版 Tangent measure In measure theory, tangent measures are used to study the local behavior of Radon measures, in much the same way as tangent spaces are used to study the local behavior of differentiable manifolds. Tangent measures (introduced by David Preiss in his study of rectifiable sets) are a useful tool in geometric measure theory. For example, they are used in proving Marstrand’s theorem and Preiss' theorem. ==Definition== Consider a Radon measure ''μ'' defined on an open subset Ω of ''n''-dimensional Euclidean space R''n'' and let ''a'' be an arbitrary point in Ω. We can “zoom in” on a small open ball of radius ''r'' around ''a'', ''B''''r''(''a''), via the transformation : $T\_(x)=\backslash frac,$ which enlarges the ball of radius ''r'' about ''a'' to a ball of radius 1 centered at 0. With this, we may now zoom in on how ''μ'' behaves on ''B''''r''(''a'') by looking at the push-forward measure defined by :$T\_\backslash mu(A)=\backslash mu(a+rA)$ where :$a+rA=\backslash .$ As ''r'' gets smaller, this transformation on the measure ''μ'' spreads out and enlarges the portion of ''μ'' supported around the point ''a''. We can get information about our measure around ''a'' by looking at what these measures tend to look like in the limit as ''r'' approaches zero. :Definition. A ''tangent measure'' of a Radon measure ''μ'' at the point ''a'' is a second Radon measure ''ν'' such that there exist sequences of positive numbers ''c''''i'' > 0 and decreasing radii ''r''''i'' → 0 such that :: $\backslash lim\_\; c\_T\_\backslash mu\; =\backslash nu$ : where the limit is taken in the weak-∗ topology, i.e., for any continuous function ''φ'' with compact support in Ω, :: $\backslash lim\_\backslash int\_\; \backslash varphi\; \backslash ,\; \backslash mathrm\; (c\_T\_\backslash mu)=\backslash int\_\; \backslash varphi\; \backslash ,\; \backslash mathrm\; \backslash nu.$ :We denote the set of tangent measures of ''μ'' at ''a'' by Tan(''μ'', ''a''). 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「Tangent 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.