The Lebesgue integral of a nonnegative measurable function is the supremum of the integrals of nonnegative simple functions below it. Integrable real and complex functions are then defined from their positive, negative, real, and imaginary parts.
A simple function is a measurable function with finite range. A nonnegative one can be written as for disjoint measurable sets , and its integral is .
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"More complex and general" integral. Matches the Riemann integral for "simple functions", but also works for some "funkier" functions that Riemann does not work for.
Ciro Santilli sometimes wonders how much someone can gain from learning this besides the beauty of mathematics, since we can hand-wave a Lebesgue integral on almost anything that is of practical use. The beauty is good reason enough though.