Absolutely integrable function

A function is said to be absolutely integrable if its 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. If we are using 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".

References

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