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 $S_n=\eta_i \mu(E_i)$ where $\eta_i$ is the value of the function in subinterval $i$, and $\mu(E_i)$ is the Lebesgue Measure of the Set $E_i$ of points for which values are approximately $\eta_i$. This type of integral covers a wider class of functions than does the Riemann Integral.

See also A-Integrable, Complete Functions, Integral

References

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.