Measure and integration The Lebesgue integral Concept
Integral of a non-negative function
Summary
The integral of a non-negative measurable function on a measure space is the supremum of the integrals of the simple functions that lie below it. It always exists, as a number in . It is the first step of the Lebesgue integral, and the monotone convergence theorem and Fatou's lemma are statements about it.
Where this leads
38 pages build on this one.
- Abelian theorems for integral transforms
- Fourier transform on Schwartz space
- Radon–Nikodym theorem
- Discrete series representation
- Oscillator representation
- Hardy's uncertainty principle
- Scheffé's lemma
- Abscissa of absolute convergence
- Smoothing by convolution
- Lp space
- Cantor–Lebesgue theorem
- Derivative of a convolution
- Numerical integration
- Dominated convergence theorem
- Vitali convergence theorem
- Entropic uncertainty principle
- Integral operator
- Radon–Nikodym derivative
- Bochner integral
- Heisenberg inequality for the Fourier transform
- Babenko–Beckner inequality
- Uniform integrability
- Monotone convergence theorem
- Layer cake representation
- Differentiation and multiplication under the Fourier transform
- Laplace transform tends to zero at infinity
- Cantor's uniqueness theorem for trigonometric series
- Integral transform
- Integrable function
- Fatou's lemma
- Measure with density
- Lebesgue integral
- Fisher–Neyman factorization theorem
- Symmetric and self-reciprocal kernels
- Stone–von Neumann theorem
- Hartogs lemma for subharmonic functions
- Fourier transform of the Gaussian
- Absolute continuity of the Lebesgue integral
Further reading
About this page
| Tag | DG76G5Z6XW |
|---|---|
| URI | m2://DG76G5Z6XW |
| Kind | concept |
| Writing | W0 stub |
| MSC | 28A25 |
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Used by
- Radon–Nikodym theorem
- Lp space
- Bochner integral
- Uniform integrability
- Monotone convergence theorem
- Layer cake representation
- Integrable function
- Fatou's lemma
- Measure with density
- Lebesgue integral
- Absolute continuity of the Lebesgue integral