## Lebesgue Integral

The Lebesgue Integral is defined in terms of upper and lower bounds using the Lebesgue Measure of a Set. It uses a Lebesgue Sum where is the value of the function in subinterval , and is the Lebesgue Measure of the Set of points for which values are approximately . This type of integral covers a wider class of functions than does the Riemann Integral.

Kestelman, H. Lebesgue Integral of a Non-Negative Function'' and Lebesgue Integrals of Functions Which Are Sometimes Negative.'' Chs. 5-6 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 113-160, 1960.