Lebesgue Integrable

A nonnegative measurable function f is called Lebesgue integrable if its Lebesgue integral intfdmu 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)^(infty)intf_n<infty.

2. f(x)=sum_(n=1)^(infty)f_n(x) almost everywhere.

See also

Integral, Lebesgue Integral, Riemann Integral, Simple Function, Step Function

This entry contributed by John Renze

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Royden, H. L. §11.3 in Real Analysis, 3rd ed. New York: Macmillan, p. 31, 1988.

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Lebesgue Integrable

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Renze, John. "Lebesgue Integrable." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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