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Absolutely integrable function : ウィキペディア英語版
Absolutely integrable function

An absolutely integrable function is a function whose absolute value is integrable, meaning that the integral of the absolute value over the whole domain is finite.
For a real-valued function, since
:\int|f|=\int f^++\int f^-
where
:f^+ = \max (f,0), \ \ \ f^- = \max(-f,0),
this implies that both \int f^+ and \int f^- must be finite. In Lebesgue integration, this is exactly the requirement for ''f'' itself to be considered integrable (with the integral then equaling \int f^+-\int f^-), so that in fact "absolutely integrable" means the same thing as "Lebesgue integrable".
The same thing goes for a complex-valued function. Having a finite integral of the absolute value is equivalent to the conditions for the function to be "Lebesgue integrable".
==External links==

*(【引用サイトリンク】title=Absolutely integrable function – Encyclopedia of Mathematics )

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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