Theory of integrals

Japanese: 積分論 - せきぶんろん（英語表記）theory of integral

Modern integral theory is constructed away from the concepts of spatial coordinates and topology, but it is an abstraction of the Lebesgue integral in Euclidean space. The Lebesgue measure in Euclidean space is an extension of the concepts of area and volume, and the empty set and the entire space are clearly measurable sets (sets for which the Lebesgue measure is defined), and any set obtained by performing the operations of union, subtraction, and intersection of measurable sets countably infinite times is also a measurable set. Furthermore, if the Lebesgue measure of a measurable set E is written as m ( E ), then for a countably infinite sequence of measurable sets { En _} , where no two of the sets intersect with each other, the following holds:

Source: Heibonsha World Encyclopedia, 2nd Edition Information

Japanese:

現代の積分論は空間の座標や位相の概念から離れて構成されるが，これはユークリッド空間におけるルベーグ積分の抽象化である。ユークリッド空間におけるルベーグ測度は面積や体積の概念の拡張であって，空集合や全空間は明らかに可測集合(ルベーグ測度が定義される集合)であり，可測集合の和，差，交わりを作る操作をたかだか可算無限回行って得られる集合は可測集合である。また可測集合Eのルベーグ測度をm(E)と書くと，どの二つも互いに交わらない可測集合の可算無限列｛E_n｝に対して，が成り立つ。

出典　株式会社平凡社世界大百科事典第２版について　情報

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