Measure and integration Measurable functions Concept
Measurable function
Summary
A measurable function is a map between measurable spaces under which the preimage of every measurable set is measurable. These are the functions that one can integrate in Lebesgue's theory, and in probability they are the random variables. Sums, products, limits and suprema of sequences of real measurable functions are again measurable.
Where this leads
73 pages build on this one.
- Abelian theorems for integral transforms
- Fourier transform on Schwartz space
- Pettis measurability theorem
- Radon–Nikodym theorem
- Measure-preserving transformation
- Schur test
- Discrete series representation
- Slowly varying function
- Oscillator representation
- Hardy's uncertainty principle
- Scheffé's lemma
- Young's inequality for integral operators
- Smoothing by convolution
- Regular variation
- Strichartz estimates in Lorentz space
- Egorov's theorem
- Lp space
- L-infinity space
- Weak Lp space
- Admissible control
- Cantor–Lebesgue theorem
- Choquet integral
- Derivative of a convolution
- Nemytskii operator
- Numerical integration
- Bartle–Dunford–Schwartz integral
- Lorentz space
- Convergence in measure
- Dominated convergence theorem
- Vitali convergence theorem
- Entropic uncertainty principle
- Quasi-measure-preserving map
- Integral operator
- Littlewood's three principles
- Radon–Nikodym derivative
- Heisenberg inequality for the Fourier transform
- Stochastic difference equation
- Square-integrable kernel
- Babenko–Beckner inequality
- Monotone convergence theorem
- Integral of a non-negative function
- Lusin's theorem
- Differentiation and multiplication under the Fourier transform
- Invariant measure
- Prékopa–Leindler inequality
- Weak-type estimate
- Riesz's subsequence theorem
- Young measure
- Product sigma-algebra
- Random integral equation
- Laplace transform tends to zero at infinity
- Disintegration theorem
- Distribution function of a measurable function
- Cantor's uniqueness theorem for trigonometric series
- Simple function
- Coarea formula
- Integrable function
- Fatou's lemma
- Measurable solutions of Cauchy's equation
- Markov kernel
- Measure with density
- Jensen inequality as a functional inequality
- Sierpiński's theorem on measurable midpoint convex functions
- Symmetric decreasing rearrangement
- Fisher–Neyman factorization theorem
- Symmetric and self-reciprocal kernels
- Stone–von Neumann theorem
- Hartogs lemma for subharmonic functions
- Orlicz space
- Fourier transform of the Gaussian
- Pushforward measure
- Essential supremum
- Absolute continuity of the Lebesgue integral
Further reading
About this page
| Tag | YYNCJSR0PJ |
|---|---|
| URI | m2://YYNCJSR0PJ |
| Kind | concept |
| Writing | W0 stub |
| MSC | 28A20 |
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Used by
- Pettis measurability theorem
- Radon–Nikodym theorem
- Measure-preserving transformation
- Schur test
- Slowly varying function
- Young's inequality for integral operators
- Regular variation
- Egorov's theorem
- Lp space
- L-infinity space
- Weak Lp space
- Admissible control
- Choquet integral
- Nemytskii operator
- Bartle–Dunford–Schwartz integral
- Lorentz space
- Convergence in measure
- Quasi-measure-preserving map
- Littlewood's three principles
- Stochastic difference equation
- Square-integrable kernel
- Monotone convergence theorem
- Integral of a non-negative function
- Lusin's theorem
- Invariant measure
- Prékopa–Leindler inequality
- Weak-type estimate
- Young measure
- Product sigma-algebra
- Random integral equation
- Disintegration theorem
- Distribution function of a measurable function
- Simple function
- Coarea formula
- Integrable function
- Measurable solutions of Cauchy's equation
- Markov kernel
- Measure with density
- Sierpiński's theorem on measurable midpoint convex functions
- Symmetric decreasing rearrangement
- Orlicz space
- Pushforward measure
- Essential supremum