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

"Absolutely integrable function – Encyclopedia of Mathematics". Retrieved 9 October 2015.

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