[過去ﾛｸﾞ] ガロア第一論文と乗数イデアル他関連資料スレ6 (1002ﾚｽ)
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287(3): 2024/01/30(火)11:23 ID:0O1eEeBq(2/12) AAS
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 外部ﾘﾝｸ:en.wikipedia.org
省12
289(1): 2024/01/30(火)11:30 ID:0O1eEeBq(4/12) AAS
>>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
290(1): 2024/01/30(火)11:40 ID:0O1eEeBq(5/12) AAS
>>287-289 補足の補足
>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
> 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.”
えーと、対偶だったな？ (>_<)
「不連続の点の集合がmeasure zeroでない→Darboux integral 不可」
の対偶
「Darboux integral 可 → 不連続の点の集合がmeasure zero 」
省4
794(2): 2024/05/11(土)15:41 ID:k0FyGno+(10/16) AAS
>>287
>柳田はちょっと前
>3年生向けの
>複素関数論の授業をしていた。
ほう、詳しいね
これは、御大か
下記ですね
省13
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