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

287 ：１３２人目の素数さん：2024/01/30(火) 11:23:57.32 ID:0O1eEeBq.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] ・・
(引用終り)

＜補足＞
１）Proofで、Darboux integral https://en.wikipedia.org/wiki/Darboux_integral
　に持ち込むのが、定石のようです
　Darboux integralを見ると、"supとinf" https://manabitimes.jp/math/1140 高校数学の美しい物語 sup(上限)とinf(下限)の意味，max・minとの違い 2022/08/15
　この"supとinf"は、数学では常用の手筋ですね
２）上記”One direction”は、「不連続の点の集合がmeasure zero→Darboux integral 不可」ですね
　背理法ですね。”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.”
３）そして、測度0でなければ
　”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.”
　を導きます
　このとき、”oscillation ”https://en.wikipedia.org/wiki/Oscillation_(mathematics)
　を使って、不連続の評価をしています。（不勉強で、”oscillation ”は初見でしたが、面白いですね。定石かも）
　”For every positive ε, Let Xε be the set of points in [a, b] with oscillation of at least ε.”に続きます

つづく

288 ：１３２人目の素数さん：2024/01/30(火) 11:24:24.41 ID:0O1eEeBq.net

つづき

この後、We now prove the converse direction は、各自ご参照ください

余談ですが、ある数学者の本の奥付に、囲碁７段格とあって やりすぎと思いましたが
囲碁用語の定石＆手筋で、数学を説明すると 分かりやすい
この方の場合、囲碁も数学の役に立っているのではと思っています　 ^^)
以上

289 ：１３２人目の素数さん：2024/01/30(火) 11:30:25.36 ID:0O1eEeBq.net

>>287 訂正

２）上記”One direction”は、「不連続の点の集合がmeasure zero→Darboux integral 不可」ですね

２）上記”One direction”は、「不連続の点の集合がmeasure zeroでない→Darboux integral 不可」ですね