A nonnegative measurable function f is called Lebesgue integrable if its Lebesgue integral integral f d μ is finite. An arbitrary measurable function is integrable if f^+ and f^- are each Lebesgue integrable, where f^+ and f^- denote the positive and negative parts of f, respectively. The following equivalent characterization of Lebesgue integrable follows as a consequence of monotone convergence theorem. A nonnegative measurable function f is Lebesgue integrable iff there exists a sequence of nonnegative simple functions {f_n} such that the following two conditions are satisfied: 1. sum_(n = 1)^∞ integral f_n<∞. 2.f(x) = sum_(n = 1)^∞ f_n(x) almost everywhere.

integral | Lebesgue integral | Riemann integral | simple function | step function