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

where

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".

单词	Absolutely integrable
释义	Absolutely integrable 英语百科 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 where 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".
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