Measure and integration The Lebesgue integral Concept
Integrable function
Summary
A measurable function on a measure space is integrable when the integral of its absolute value is finite. These are the functions for which the Lebesgue integral is a well-defined finite number. Integrable functions form the space , and the convergence theorems of integration are stated for them.
Where this leads
70 pages build on this one.
- Abelian theorems for integral transforms
- Fundamental strip of a Mellin transform
- Fourier transform on Schwartz space
- Ergodic decomposition
- Mellin convergence
- Hardy's uncertainty principle
- Scheffé's lemma
- Abscissa of absolute convergence
- Radon transform
- Smoothing by convolution
- Parseval formula for the Mellin transform
- Pettis integral
- Levinson's theorem on asymptotic integration
- Fourier transform
- Hardy–Littlewood maximal function
- X-ray transform
- FBI transform
- Kingman's subadditive ergodic theorem
- Abstract Volterra equation
- Cantor–Lebesgue theorem
- Derivative of a convolution
- Fourier transform as Laplace transform on the imaginary axis
- Lebesgue point
- Absolutely convergent Fourier series
- Dominated convergence theorem
- Vitali convergence theorem
- Entropic uncertainty principle
- Lebesgue–Stieltjes integration
- Integral operator
- Differentiability of the Mellin transform
- Heisenberg inequality for the Fourier transform
- Dini test
- Babenko–Beckner inequality
- Uniform integrability
- Riemann localization principle
- Differentiation and multiplication under the Fourier transform
- Sine and cosine series
- Birkhoff's ergodic theorem
- Strong maximal function
- Fundamental lemma of the calculus of variations
- Jensen's inequality for integrals
- Inverse Mellin transform as inverse Fourier transform
- Laplace transform tends to zero at infinity
- Oseledets theorem
- Cantor's uniqueness theorem for trigonometric series
- Mellin convolution
- Inverse Mellin transform
- Sine and cosine transforms
- Integral transform
- Muckenhoupt weights
- Fundamental theorem of calculus for Lebesgue integrals
- Lebesgue differentiation theorem
- Fourier coefficient
- Fourier series on the torus
- Fourier inversion theorem
- Integral transforms of tempered distributions
- Maximal ergodic theorem
- Pesin theory
- Szegő limit theorems
- Riemann–Lebesgue lemma
- Mellin's inverse formula
- Decay of Fourier coefficients and smoothness
- Mellin inversion theorem
- Lebesgue integral
- Asymptotic expansion from poles of a Mellin transform
- Hartley transform
- Fourier transform of the Gaussian
- Inverse Fourier transform
- Mellin transform as a Fourier transform in logarithmic variables
- Absolute continuity of the Lebesgue integral
Further reading
About this page
| Tag | X7GTF3D98F |
|---|---|
| URI | m2://X7GTF3D98F |
| Kind | concept |
| Writing | W0 stub |
| MSC | 28A25, 26A42 |
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Used by
- Fundamental strip of a Mellin transform
- Mellin convergence
- Scheffé's lemma
- Radon transform
- Parseval formula for the Mellin transform
- Pettis integral
- Levinson's theorem on asymptotic integration
- Fourier transform
- Hardy–Littlewood maximal function
- X-ray transform
- FBI transform
- Kingman's subadditive ergodic theorem
- Abstract Volterra equation
- Fourier transform as Laplace transform on the imaginary axis
- Lebesgue point
- Absolutely convergent Fourier series
- Dominated convergence theorem
- Vitali convergence theorem
- Lebesgue–Stieltjes integration
- Differentiability of the Mellin transform
- Uniform integrability
- Sine and cosine series
- Birkhoff's ergodic theorem
- Strong maximal function
- Fundamental lemma of the calculus of variations
- Jensen's inequality for integrals
- Inverse Mellin transform as inverse Fourier transform
- Mellin convolution
- Inverse Mellin transform
- Sine and cosine transforms
- Muckenhoupt weights
- Fundamental theorem of calculus for Lebesgue integrals
- Lebesgue differentiation theorem
- Fourier coefficient
- Fourier series on the torus
- Fourier inversion theorem
- Maximal ergodic theorem
- Szegő limit theorems
- Riemann–Lebesgue lemma
- Mellin inversion theorem
- Lebesgue integral
- Hartley transform
- Inverse Fourier transform
- Mellin transform as a Fourier transform in logarithmic variables
- Absolute continuity of the Lebesgue integral