A nonnegative measurable function is called Lebesgue integrable if its Lebesgue integral is finite. An arbitrary measurable function is integrable if and are each Lebesgue integrable, where and denote the positive and negative parts of , respectively.

The following equivalent characterization of Lebesgue integrable follows as a consequence of monotone convergence theorem. A nonnegative measurable function is Lebesgue integrable iff there exists a sequence of nonnegative simple functions such that the following two conditions are satisfied:

1. .