In Lebesgue integration, this is exactly the requirement for any measurable function f to be considered integrable, with the integral then equaling , so that in fact "absolutely integrable" means the same thing as "Lebesgue integrable" for measurable functions.
The concept of integrability in Lebesgue integration hinges on the same idea as "absolutely integrable," which means a function must satisfy this specific condition to be considered integrable.