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# Lebesgue Integrable

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. .

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

This entry contributed by John Renze

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## References

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

## Referenced on Wolfram|Alpha

Lebesgue Integrable

## Cite this as:

Renze, John. "Lebesgue Integrable." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LebesgueIntegrable.html