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.
In Lebesgue integration, a function must meet this specific requirement to be considered integrable. The integral then equals , and thus, "absolutely integrable" is synonymous with "Lebesgue integrable" for measurable functions.