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| differentiation of integrals ： ウィキペディア英語版 |
differentiation of integrals
In mathematics, the problem of differentiation of integrals is that of determining under what circumstances the mean value integral of a suitable function on a small neighbourhood of a point approximates the value of the function at that point. More formally, given a space ''X'' with a measure ''μ'' and a metric ''d'', one asks for what functions ''f'' : ''X'' → R does
:
for all (or at least ''μ''-almost all) ''x'' ∈ ''X''? (Here, as in the rest of the article, ''B''''r''(''x'') denotes the open ball in ''X'' with ''d''-radius ''r'' and centre ''x''.) This is a natural question to ask, especially in view of the heuristic construction of the Riemann integral, in which it is almost implicit that ''f''(''x'') is a "good representative" for the values of ''f'' near ''x''.
== Theorems on the differentiation of integrals ==
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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