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 ∫ f + and ∫ f − must be finite. In Lebesgue integration, this is exactly the requirement for f itself to be considered integrable (with the integral then equaling ∫ f + − ∫ 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".