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}.
Analytic-function sequence limits are integrals {Stieltjes integral, Riemann}. Stieltjes integrals generalize simpler integrals {Riemann-Darboux integral}.
Integrals, from x = a to x = b, of f(x) * dg(x) * dx equal limits of sums, from i = 0 to i = n, of f(e(i)) * (g(x(i + 1)) - g(x(i))), where x(i) are partition intervals and e(i) are inside intervals (x(i), x(i + 1)).
Riemann
Functions {Riemann integrable function} can have no discontinuities or have discontinuities that form measure-zero sets. Riemann integrals are Lebesgue integrable, but Lebesgue integrals can be not Riemann integrable.
Analytic-function sequence limits are integrals {Stieltjes integral}. Stieltjes integrals are generalized Riemann-Darboux integrals. Integrals, from x = a to x = b, of f(x) * dg(x) * dx equals limits of sums, from i = 0 to i = n, of f(e(i)) * (g(x(i+1)) - g(x(i))), where x(i) are partition intervals and e(i) are inside intervals (x(i), x(i+1)).
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Date Modified: 2022.0225