逆に、リーマン積分を勉強する意味はあるのか

[0001 １３２人目の素数さん 2022/10/26(水) 13:04:59.53 ID:vqY6TGmd]

ルベーグ積分でいいんじゃないか

[0002 １３２人目の素数さん 2022/10/26(水) 16:09:29.98 ID:e6Te0RVI]

リーマン積分でないと成り立たない定理が微妙に存在するんだよな。

https://en.wikipedia.org/wiki/Equidistributed_sequence

＞Suppose (s1, s2, s3, ...) is a sequence contained in the interval [a, b].
＞Then the following conditions are equivalent:
＞
＞1. The sequence is equidistributed on [a, b].
＞2. For every Riemann-integrable (complex-valued) function f : [a, b] → C, the following limit holds:
＞ lim[N→∞] (1/N)Σ[n＝1〜N]f(s_n)＝1/(b−a)∫[a,b]f(x)dx

＞It is not possible to generalize the integral criterion to a class of functions
＞bigger than just the Riemann-integrable ones. For example, if the Lebesgue integral is considered and
＞f is taken to be in L1, then this criterion fails. As a counterexample, take f to be the indicator function of
＞some equidistributed sequence. Then in the criterion, the left hand side is always 1,
＞whereas the right hand side is zero, because the sequence is countable, so f is zero almost everywhere.

[0003 １３２人目の素数さん 2022/10/26(水) 17:43:48.79 ID:yQGb1mps]

正負に値が振動する無限区間の積分などは、ルベーグではどうだったかな。