Measure and integration Construction of measures Concept
Outer measure
Summary
An outer measure assigns a size in to every subset of a set, not only to the sets of a -algebra. It is monotone and countably subadditive, but in general not additive. Through Carathéodory's criterion every outer measure produces a -algebra on which it is a complete measure. This is the standard way to build Lebesgue measure and Hausdorff measures.
Where this leads
105 pages build on this one.
- Baker's map
- Interval exchange transformation
- Nonexistence of infinite-dimensional Lebesgue measure
- Koksma–Hlawka inequality
- Irrational rotation
- Lebesgue measure
- Transfer operator
- Feigenbaum–Cvitanović functional equation
- Standard probability space
- Milnor–Thurston kneading theory
- Minkowski content
- Benedicks' theorem
- Poincaré classification of circle homeomorphisms
- Circle map
- Vanishing mean oscillation
- Double bubble theorem
- Kolmogorov's weak-type estimate for the conjugate function
- Hardy–Littlewood maximal function
- Cantor–Lebesgue theorem
- Brunn–Minkowski theorem
- Minkowski's first inequality
- Mixed volume
- Liouville's theorem on phase-space volume
- Hausdorff measure
- Crisis
- Blichfeldt's theorem
- Packing density
- Calderón–Zygmund theorem
- Atomic decomposition of Hardy spaces
- Lebesgue point
- Rademacher functions
- Hardy–Littlewood maximal inequality
- Arnold's cat map
- Covering density
- Tent map
- Monte Carlo integration
- Littlewood's three principles
- Faber–Krahn inequality
- John ellipsoid
- Duistermaat–Heckman formula
- Steinhaus theorem
- Feigenbaum constants
- Intermittency
- Dini test
- Pólya's conjecture on Laplacian eigenvalues
- Dido's problem
- Whitney decomposition
- Mahler conjecture
- Hahn–Kolmogorov theorem
- Ham sandwich theorem
- Lusin's theorem
- Layer cake representation
- Riemann localization principle
- Hardy–Littlewood–Sobolev inequality
- Bifurcation diagram
- Dyadic transformation
- Smith–Volterra–Cantor set
- Carathéodory's criterion
- Strong maximal function
- Young's convolution inequality
- Banach measure problem
- Calderón–Zygmund lemma
- Poincaré–Birkhoff theorem
- Isoperimetric inequality
- Pompeiu problem
- Danzer set
- Blaschke–Santaló inequality
- Gauss map
- Busemann–Petty problem
- Perimeter of a set
- Hermite–Genocchi formula
- Fefferman duality theorem
- Cantor's uniqueness theorem for trigonometric series
- Gaussian measure
- Denjoy theorem on rotation number
- Muckenhoupt weights
- Coarea formula
- Logistic map
- Packing measure
- John–Nirenberg inequality
- Fundamental theorem of calculus for Lebesgue integrals
- Lebesgue differentiation theorem
- Commutativity of convolution
- Fractional integration
- Löwner–John theorem
- Measurable solutions of Cauchy's equation
- Measure problem of Lebesgue
- Caccioppoli set
- Masur–Veech theorem
- Brunn–Minkowski theory
- Jakobson's theorem
- Besicovitch set of measure zero
- Lebesgue's density theorem
- Riemann–Lebesgue lemma
- Vitali set
- Kakeya needle problem
- Pseudo-Anosov map
- Jensen inequality as a functional inequality
- Metric outer measure
- Decay of Fourier coefficients and smoothness
- Singer's theorem
- Sierpiński's theorem on measurable midpoint convex functions
- Isomorphism theorem for measure algebras
- Symmetric decreasing rearrangement
- Riesz potential operator
Further reading
About this page
| Tag | BJ314H17HC |
|---|---|
| URI | m2://BJ314H17HC |
| Kind | concept |
| Writing | W0 stub |
| MSC | 28A12 |
| Same as |
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Used by
- Lebesgue measure
- Hausdorff measure
- Hahn–Kolmogorov theorem
- Carathéodory's criterion
- Packing measure
- Metric outer measure