Functions {Lebesgue integral} can have sums of lengths over intervals.
purposes
Lebesgue integrals can integrate discontinuous functions.
finite
Lebesgue integrals can be finite {summable function}. Limit from x = a to x = b of f(x) * cos(n*x) * dx equals zero. Limit from x = a to x = b of f(x) * sin(n*x) * dx equals zero. Therefore, Lebesgue integral can use Fourier series {Riemann-Lebesgue lemma}.
finite: convergence
Functions can have no bound in interval, but Lebesgue integral can converge absolutely.
extensions
Lebesgue-integral extensions include spectral theory {Lebesgue-Stieltjes integral} {ergodic theory} {harmonic analysis} {generalized Fourier analysis}.
Mathematical Sciences>Calculus>Analysis>Integral
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Date Modified: 2022.0224