integrable function

Integrating non-negative functions always gives an answer in [0, infinity], but for a function that goes both above and below the axis we want a finite, signed number. The clean rule is: a function is integrable exactly when there is no infinite cancellation hiding in it — when the total area, ignoring sign, is finite. Demand that the integral of the absolute value be finite, and everything else falls into place.

Precisely, a measurable function f (real or complex) is Lebesgue integrable if the integral of |f| dμ is finite. Then write f as f-plus minus f-minus (its positive and negative parts), both non-negative with finite integral, and define the integral of f as the integral of f-plus minus the integral of f-minus. For complex f, integrate the real and imaginary parts separately. This is consistent and gives a finite value.

Note the asymmetry with conditionally convergent series and improper Riemann integrals: Lebesgue integrability is an absolute notion. There is no Lebesgue analogue of a conditionally convergent integral; if |f| has infinite integral, f is simply not integrable, even if a delicate Riemann-style limit of (sin x)/x would converge. The payoff is robustness: integrable functions form a vector space and obey the powerful convergence theorems.