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.
See also A-Integrable, Complete Functions, 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.