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

both ${\displaystyle \int f^{+}(x)dx}$ and ${\displaystyle \int f^{-}(x)dx}$ must be finite. In Lebesgue integration, this is exactly the requirement for f itself to be considered integrable (with the integral then equaling ${\displaystyle \int f^{+}(x)dx-\int f^{-}(x)dx}$), so that in fact "absolutely integrable" means the same thing as "Lebesgue integrable".

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