翻訳と辞書 Words near each other ・ Leberecht Uhlich ・ Leberknödel ・ Leberkäse ・ Lebern District ・ Lebert ・ Lebert Lombardo ・ Lebes ・ Lebes Gamikos ・ Lebesby ・ Lebesby (village) ・ Lebesby Church ・ Lebesgue (crater) ・ Lebesgue constant ・ Lebesgue constant (interpolation) ・ Lebesgue covering dimension ・ Lebesgue differentiation theorem ・ Lebesgue integrability ・ Lebesgue integration ・ Lebesgue measure ・ Lebesgue point ・ Lebesgue space ・ Lebesgue spine ・ Lebesgue's decomposition theorem ・ Lebesgue's density theorem ・ Lebesgue's lemma ・ Lebesgue's number lemma ・ Lebesgue–Stieltjes integration ・ Lebesque ・ Lebet ・ Lebetain Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Lebesgue differentiation theorem ： ウィキペディア英語版 Lebesgue differentiation theorem In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point. The theorem is named for Henri Lebesgue. ==Statement== For a Lebesgue integrable real or complex-valued function ''f'' on R''n'', the indefinite integral is a set function which maps a measurable set ''A''  to the Lebesgue integral of $f\; \backslash cdot\; \backslash mathbf\_A$, where $\backslash mathbf\_$ denotes the characteristic function of the set ''A''. It is usually written :: $A\; \backslash mapsto\; \backslash int\_f\backslash \; \backslash mathrm\backslash lambda,$ with ''λ'' the ''n''–dimensional Lebesgue measure. The ''derivative'' of this integral at ''x'' is defined to be ::$\backslash lim\_\; \backslash frac\; \backslash int\_f\; \backslash ,\; \backslash mathrm\backslash lambda,$ where |''B''| denotes the volume (''i.e.'', the Lebesgue measure) of a ball ''B''  centered at ''x'', and ''B'' → ''x'' means that the diameter of ''B''  tends to 0. The ''Lebesgue differentiation theorem'' states that this derivative exists and is equal to ''f''(''x'') at almost every point ''x'' ∈ R''n''. In fact a slightly stronger statement is true. Note that: ::$\backslash left|\backslash frac\; \backslash int\_f(y)\; \backslash ,\; \backslash mathrm\backslash lambda(y)\; -\; f(x)\backslash right|\; =\; \backslash left|\backslash frac\; \backslash int\_f(y)\; -f(x)\backslash ,\; \backslash mathrm\backslash lambda(y)\backslash right|\; \backslash le\; \backslash frac\; \backslash int\_|f(y)\; -f(x)|\backslash ,\; \backslash mathrm\backslash lambda(y).$ The stronger assertion is that the right hand side tends to zero for almost every point ''x''. The points ''x'' for which this is true are called the Lebesgue points of ''f''. A more general version also holds. One may replace the balls ''B''  by a family $\backslash mathcal$ of sets ''U''  of ''bounded eccentricity''. This means that there exists some fixed ''c'' > 0 such that each set ''U''  from the family is contained in a ball ''B''  with $|U|\; \backslash ge\; c\; \backslash ,\; |B|$. It is also assumed that every point ''x'' ∈ R''n'' is contained in arbitrarily small sets from $\backslash mathcal$. When these sets shrink to ''x'', the same result holds: for almost every point ''x'', ::$f(x)\; =\; \backslash lim\_\; \backslash int\_U\; f\; \backslash ,\; \backslash mathrm\backslash lambda.$ The family of cubes is an example of such a family $\backslash mathcal$, as is the family $\backslash mathcal$(''m'') of rectangles in R2 such that the ratio of sides stays between ''m''−1 and ''m'', for some fixed ''m'' ≥ 1. If an arbitrary norm is given on R''n'', the family of balls for the metric associated to the norm is another example. The one-dimensional case was proved earlier by . If ''f'' is integrable on the real line, the function :$F(x)\; =\; \backslash int\_^x\; f(t)\; \backslash ,\; \backslash mathrm\; t$ is almost everywhere differentiable, with $F\text{'}(x)\; =\; f(x).$ 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「Lebesgue differentiation theorem」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.