ガロア第一論文と乗数イデアル他関連資料スレ6

1 ：１３２人目の素数さん：2024/01/08(月) 09:09:43.45 ID:OXe7qSh4.net

このスレは、ガロア第一論文と乗数イデアル他関連資料スレです
関連は、だいたい何でもありです（現代ガロア理論＆乗数イデアル関連他文学論まで）

前スレ
ガロア第一論文と乗数イデアル他関連資料スレ5
https://rio2016.5ch.net/test/read.cgi/math/1687778456/

資料としては、まずはこれ
https://sites.google.com/site/galois1811to1832/
ガロアの第一論文を読む
渡部 一己 著 （2018.1.28）
PDF
https://sites.google.com/site/galois1811to1832/galois-1.pdf?attredirects=0

＜乗数イデアル関連＞
ガロア第一論文及びその関連の資料スレ
https://rio2016.5ch.net/test/read.cgi/math/1615510393/785 以降ご参照
https://en.wikipedia.org/wiki/Multiplier_ideal Multiplier ideal
https://mathoverflow.net/questions/142937/motivation-for-multiplier-ideal-sheaves motivation for multiplier ideal sheaves asked Sep 23, 2013 Koushik

＜層について＞
https://ja.wikipedia.org/wiki/%E5%B1%A4_(%E6%95%B0%E5%AD%A6)
層 (数学)
https://en.wikipedia.org/wiki/Sheaf_(mathematics)
Sheaf (mathematics)
https://fr.wikipedia.org/wiki/Faisceau_(math%C3%A9matiques)
Faisceau (mathématiques)

あと、テンプレ順次

つづく

266 ：１３２人目の素数さん：2024/01/29(月) 07:58:17.77 ID:2Tor3z84.net

>>261-263
>>>260 その指摘が不明確かと

やれやれ、日wikipediaに書いてあることを自慢して
ちょっとツッコミあると沈没か？ｗ

さて
リーマン可積分⇒微分可能でない点の集合が測度０
　↓
リーマン可積分⇒連続でない点の集合が測度０
ですな

"R^n の有界閉区間 I 上の有界関数 f: I → R に対し、
f が I 上リーマン可積分であることと、
f がほとんど至るところ連続であること（※）は同値
（※ f の不連続点全体の集合が零集合）"
だったね

これの証明は、下記の英wikipediaにある
（Integrability 可積分性条件 the Lebesgue-Vitali theorem な）
(参考)
https://en.wikipedia.org/wiki/Riemann_integral
Riemann integral

Integrability
A bounded function on a compact interval [a, b] is Riemann integrable if and only if it is continuous almost everywhere (the set of its points of discontinuity has measure zero, in the sense of Lebesgue measure). This is the Lebesgue-Vitali theorem (of characterization of the Riemann integrable functions). It has been proven independently by Giuseppe Vitali and by Henri Lebesgue in 1907, and uses the notion of measure zero, but makes use of neither Lebesgue's general measure or integral.
The integrability condition can be proven in various ways,[4][5][6][7] one of which is sketched below.

Proof
The proof is easiest using the Darboux integral definition of integrability (formally, the Riemann condition for integrability) – a function is Riemann integrable if and only if the upper and lower sums can be made arbitrarily close by choosing an appropriate partition.
One direction can be proven using the oscillation definition of continuity:[8] For every positive ε, Let Xε be the set of points in [a, b] with oscillation of at least ε. Since every point where f is discontinuous has a positive oscillation and vice versa, the set of points in [a, b], where f is discontinuous is equal to the union over {X1/n} for all natural numbers n.

つづく

267 ：１３２人目の素数さん：2024/01/29(月) 07:58:35.71 ID:2Tor3z84.net

つづき

If this set does not have zero Lebesgue measure, then by countable additivity of the measure there is at least one such n so that X1/n does not have a zero measure. Thus there is some positive number c such that every countable collection of open intervals covering X1/n has a total length of at least c. In particular this is also true for every such finite collection of intervals. This remains true also for X1/n less a finite number of points (as a finite number of points can always be covered by a finite collection of intervals with arbitrarily small total length).

For every partition of [a, b], consider the set of intervals whose interiors include points from X1/n. These interiors consist of a finite open cover of X1/n, possibly up to a finite number of points (which may fall on interval edges). Thus these intervals have a total length of at least c. Since in these points f has oscillation of at least 1/n, the infimum and supremum of f in each of these intervals differ by at least 1/n. Thus the upper and lower sums of f differ by at least c/n. Since this is true for every partition, f is not Riemann integrable.

We now prove the converse direction using the sets Xε defined above.[9] For every ε, Xε is compact, as it is bounded (by a and b) and closed:

For every series of points in Xε that is converging in [a, b], its limit is in Xε as well. This is because every neighborhood of the limit point is also a neighborhood of some point in Xε, and thus f has an oscillation of at least ε on it. Hence the limit point is in Xε.
Now, suppose that f is continuous almost everywhere. Then for every ε, Xε has zero Lebesgue measure. Therefore, there is a countable collections of open intervals in [a, b] which is an open cover of Xε, such that the sum over all their lengths is arbitrarily small. Since Xε is compact, there is a finite subcover – a finite collections of open intervals in [a, b] with arbitrarily small total length that together contain all points in Xε. We denote these intervals {I(ε)i}, for 1 ≤ i ≤ k, for some natural k.

The complement of the union of these intervals is itself a union of a finite number of intervals, which we denote {J(ε)i} (for 1 ≤ i ≤ k − 1 and possibly for i = k, k + 1 as well).

つづく

268 ：１３２人目の素数さん：2024/01/29(月) 07:58:50.16 ID:2Tor3z84.net

つづき

We now show that for every ε > 0, there are upper and lower sums whose difference is less than ε, from which Riemann integrability follows. To this end, we construct a partition of [a, b] as follows:

Denote ε1 = ε / 2(b − a) and ε2 = ε / 2(M − m), where m and M are the infimum and supremum of f on [a, b]. Since we may choose intervals {I(ε1)i} with arbitrarily small total length, we choose them to have total length smaller than ε2.

Each of the intervals {J(ε1)i} has an empty intersection with Xε1, so each point in it has a neighborhood with oscillation smaller than ε1. These neighborhoods consist of an open cover of the interval, and since the interval is compact there is a finite subcover of them. This subcover is a finite collection of open intervals, which are subintervals of J(ε1)i (except for those that include an edge point, for which we only take their intersection with J(ε1)i). We take the edge points of the subintervals for all J(ε1)i − s, including the edge points of the intervals themselves, as our partition.

Thus the partition divides [a, b] to two kinds of intervals:

Intervals of the latter kind (themselves subintervals of some J(ε1)i). In each of these, f oscillates by less than ε1. Since the total length of these is not larger than b − a, they together contribute at most ε∗
1(b − a) = ε/2 to the difference between the upper and lower sums of the partition.
The intervals {I(ε)i}. These have total length smaller than ε2, and f oscillates on them by no more than M − m. Thus together they contribute less than ε∗
2(M − m) = ε/2 to the difference between the upper and lower sums of the partition.
In total, the difference between the upper and lower sums of the partition is smaller than ε, as required.
(引用終り)
以上

269 ：１３２人目の素数さん：2024/01/29(月) 08:07:05.92 ID:2Tor3z84.net

補足
>Proof
>The proof is easiest using the Darboux integral definition of integrability (formally, the Riemann condition for integrability) – a function is Riemann integrable if and only if the upper and lower sums can be made arbitrarily close by choosing an appropriate partition.

”the Darboux integral definition of integrability”は、日wikipediaにも説明あるよ
もちろん、英wikipediaにも詳しい説明がある（下記）

(参考)
https://ja.wikipedia.org/wiki/%E3%83%AA%E3%83%BC%E3%83%9E%E3%83%B3%E7%A9%8D%E5%88%86
リーマン積分

類似概念
リーマン積分の定義によく用いられるのがダルブー積分である。これは、ダルブー積分が技術的に単純で、リーマン可積分性とダルブー可積分性が同値になることによる。

https://en.wikipedia.org/wiki/Darboux_integral
Darboux integral

In the branch of mathematics known as real analysis, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal.[1] The definition of the Darboux integral has the advantage of being easier to apply in computations or proofs than that of the Riemann integral. Consequently, introductory textbooks on calculus and real analysis often develop Riemann integration using the Darboux integral, rather than the true Riemann integral.[2] Moreover, the definition is readily extended to defining Riemann–Stieltjes integration.[3] Darboux integrals are named after their inventor, Gaston Darboux (1842–1917).

Definition
略