Measure and integration The Lebesgue integral Concept
Lebesgue integral
Summary
The Lebesgue integral assigns a number to each integrable function on a measure space. It is built from simple functions, then non-negative functions, then general functions split into positive and negative parts. It extends the Riemann integral to many more functions and to any measure space. Its great strength is the convergence theorems, which let one exchange limits and integrals under mild conditions.
Where this leads
68 pages build on this one.
- Abelian theorems for integral transforms
- Fourier transform on Schwartz space
- Operational calculus
- Koksma–Hlawka inequality
- Stein's unbiased risk estimate
- Ill-posedness of real Laplace inversion
- Hardy's uncertainty principle
- Smoothing by convolution
- Pettis integral
- Fourier transform
- Kolmogorov's weak-type estimate for the conjugate function
- Torus integral
- Oscillatory integral
- Stein's example
- Singular integral operator
- Cantor–Lebesgue theorem
- Derivative of a convolution
- Calderón–Zygmund theorem
- Dominated convergence theorem
- Entropic uncertainty principle
- Lebesgue–Stieltjes integration
- Riesz–Markov–Kakutani representation theorem
- Heisenberg inequality for the Fourier transform
- Integral functional with a Lagrangian density
- Hankel transform
- Dini test
- Square-integrable kernel
- Convolution on the line
- Babenko–Beckner inequality
- Abel transform
- Bayes risk
- Strip of convergence of the bilateral transform
- Conjugate function
- Riemann localization principle
- Differentiation and multiplication under the Fourier transform
- Prékopa–Leindler inequality
- Fundamental lemma of the calculus of variations
- Jensen's inequality for integrals
- Calderón–Zygmund lemma
- Weak convergence of measures
- Existence and uniqueness of Haar measure
- Abel transform as the projection of a radial function
- Mellin transform
- Laplace transform tends to zero at infinity
- Cantor's uniqueness theorem for trigonometric series
- Mellin convolution
- Inverse Mellin transform
- Modular function
- Short-time Fourier transform
- Vanishing moments of a wavelet
- John–Nirenberg inequality
- Wasserstein metric
- Fourier coefficient
- Kantorovich–Rubinstein duality
- Fourier–Legendre series
- Fractional integration
- Integral transforms of tempered distributions
- Lerch's uniqueness theorem
- Two-sided Laplace transform
- Convolution on the half-line
- Weak derivative
- Riemann–Lebesgue lemma
- Decay of Fourier coefficients and smoothness
- Circle average
- Orlicz space
- Kontorovich–Lebedev transform
- Fourier transform of the Gaussian
- Inverse Fourier transform
Further reading
About this page
| Tag | DWJF7ABPR1 |
|---|---|
| URI | m2://DWJF7ABPR1 |
| Kind | concept |
| Writing | W0 stub |
| MSC | 28A25, 26A42 |
| Same as |
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What links here
Used by
- Koksma–Hlawka inequality
- Ill-posedness of real Laplace inversion
- Pettis integral
- Fourier transform
- Torus integral
- Oscillatory integral
- Singular integral operator
- Dominated convergence theorem
- Lebesgue–Stieltjes integration
- Riesz–Markov–Kakutani representation theorem
- Integral functional with a Lagrangian density
- Hankel transform
- Square-integrable kernel
- Convolution on the line
- Abel transform
- Bayes risk
- Strip of convergence of the bilateral transform
- Conjugate function
- Prékopa–Leindler inequality
- Fundamental lemma of the calculus of variations
- Jensen's inequality for integrals
- Calderón–Zygmund lemma
- Weak convergence of measures
- Abel transform as the projection of a radial function
- Mellin transform
- Mellin convolution
- Inverse Mellin transform
- Short-time Fourier transform
- Vanishing moments of a wavelet
- Wasserstein metric
- Fourier coefficient
- Kantorovich–Rubinstein duality
- Fourier–Legendre series
- Fractional integration
- Lerch's uniqueness theorem
- Two-sided Laplace transform
- Convolution on the half-line
- Weak derivative
- Riemann–Lebesgue lemma
- Circle average
- Orlicz space
- Kontorovich–Lebedev transform
- Inverse Fourier transform
Results about this
- Dominated convergence theorem
- Vitali convergence theorem
- Prékopa–Leindler inequality
- Jensen's inequality for integrals