[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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267(2): 2024/01/29(月)07:58 ID:2Tor3z84(2/4) 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] 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.
省2
270(1): 2024/01/29(月)09:37 ID:2GVFwqXV(1/2) AAS
>>266-269 それ日本で要約して書いてみ
271(4): 2024/01/29(月)10:41 ID:F9Ii6wqO(1) AAS
>>270
>>>266-269 それ日本で要約して書いてみ
面白い漫才師だな
するりとうまく「体を入れ替える」話法ねｗ
下記は、君が”大学数学ゼミ、かくあるべし”！>>262
と主張していたことだよ
で、そのしたり顔の主張をつぶしに行ったのが、私です
省27
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