ガロア第一論文と乗数イデアル他関連資料スレ6

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

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

http://rio2016.5ch.net/test/read.cgi/math/1704672583/269

482: １３２人目の素数さん [] 2024/02/06(火) 16:30:40.92 ID:waUghugl

つづき

(参考)
https://en.wikipedia.org/wiki/Peano%E2%80%93Jordan_measure
In mathematics, the Peano–Jordan measure (also known as the Jordan content) is an extension of the notion of size (length, area, volume) to shapes more complicated than, for example, a triangle, disk, or parallelepiped.
Extension to more complicated sets
For example, the fat Cantor set is not. Its inner Jordan measure vanishes, since its complement is dense; however, its outer Jordan measure does not vanish, since it cannot be less than (in fact, is equal to) its Lebesgue measure. Also, a bounded open set is not necessarily Jordan measurable. For example, the complement of the fat Cantor set (within the interval) is not.

A bounded set is Jordan measurable if and only if its indicator function is Riemann-integrable, and the value of the integral is its Jordan measure.[1]

Equivalently, for a bounded set B the inner Jordan measure of B is the Lebesgue measure of the topological interior of B and the outer Jordan measure is the Lebesgue measure of the closure.[4]
From this it follows that a bounded set is Jordan measurable if and only if its topological boundary has Lebesgue measure zero. (Or equivalently, if the boundary has Jordan measure zero; the equivalence holds due to compactness of the boundary.)

References
[1] While a set whose measure is defined is termed measurable, there is no commonly accepted term to describe a set whose Jordan content is defined. Munkres (1991) suggests the term "rectifiable" as a generalization of the use of this term to describe curves. Other authors have used terms including "admissible" (Lang, Zorich); "pavable" (Hubbard); "have content" (Burkill); "contented" (Loomis and Sternberg).

https://en.wikipedia.org/wiki/Indicator_function
Indicator function
This article is about the 0-1 indicator function. For the 0-infinity indicator function, see characteristic function (convex analysis).
For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers.

https://ja.wikipedia.org/wiki/%E6%8C%87%E7%A4%BA%E9%96%A2%E6%95%B0
指示関数
指示関数（indicator function）、集合の定義関数[1]、特性関数（characteristic function）は、集合の元がその集合の特定の部分集合に属するかどうかを指定することによって定義される関数である[注釈 1]。
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1704672583/482

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