Measure and integration Set systems Concept
Borel sigma-algebra
Summary
The Borel σ-algebra of a topological space is the smallest σ-algebra that contains all open sets. Its elements are the Borel sets. On the real line it is the usual family of events for real random variables. The node Borel sigma-algebra gives the precise definition.
Where this leads
127 pages build on this one.
- Baker's map
- Spectral theorem for unbounded self-adjoint operators
- 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
- Capacitability of Borel sets
- Circle map
- Vanishing mean oscillation
- Double bubble theorem
- Kolmogorov's weak-type estimate for the conjugate function
- Hardy–Littlewood maximal function
- Borel determinacy theorem
- Cantor–Lebesgue theorem
- Suslin's theorem
- Brunn–Minkowski theorem
- Tightness of measures
- Minkowski's first inequality
- Mixed volume
- Liouville's theorem on phase-space volume
- Hausdorff measure
- Crisis
- Blichfeldt's theorem
- Packing density
- Kantorovich duality
- Calderón–Zygmund theorem
- Atomic decomposition of Hardy spaces
- Lebesgue point
- Borel measure
- Rademacher functions
- Hardy–Littlewood maximal inequality
- Arnold's cat map
- Covering density
- Robust statistics
- Haar measure
- Brenier's theorem
- Tent map
- Standard Borel space
- Monte Carlo integration
- Stone's theorem on one-parameter unitary groups
- 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
- Ham sandwich theorem
- Lusin's theorem
- Layer cake representation
- Riemann localization principle
- Hardy–Littlewood–Sobolev inequality
- Bifurcation diagram
- Dyadic transformation
- Smith–Volterra–Cantor set
- Strong maximal function
- Fourier–Stieltjes transform
- Young's convolution inequality
- Banach measure problem
- Calderón–Zygmund lemma
- Weak convergence of measures
- Strang splitting
- Poincaré–Birkhoff theorem
- Optimal transport and the Monge problem
- Isoperimetric inequality
- Lie–Trotter product formula
- Pompeiu problem
- Borel functional calculus
- 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
- Displacement convexity
- Measurable function
- Denjoy theorem on rotation number
- Muckenhoupt weights
- Coarea formula
- Logistic map
- John–Nirenberg inequality
- Cylinder set measure
- Borel hierarchy
- Fundamental theorem of calculus for Lebesgue integrals
- Lebesgue differentiation theorem
- Commutativity of convolution
- Fractional integration
- Lévy–Prokhorov metric
- Löwner–John theorem
- Measurable solutions of Cauchy's equation
- Measure problem of Lebesgue
- Caccioppoli set
- Masur–Veech theorem
- Brunn–Minkowski theory
- De Finetti's theorem
- Jakobson's theorem
- Besicovitch set of measure zero
- Lebesgue's density theorem
- Riemann–Lebesgue lemma
- Vitali set
- Kakeya needle problem
- Pseudo-Anosov map
- Spectral theorem for bounded normal operators
- Jensen inequality as a functional inequality
- Lebesgue–Stieltjes 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
- Web https://en.wikipedia.org/wiki/Borel_set, lead section
About this page
| Tag | 0WQXY32Q8D |
|---|---|
| URI | m2://0WQXY32Q8D |
| Aliases | borel-algebra, borel-set, borel-sets |
| Kind | concept |
| Writing | W1 stated |
| MSC | 28A05 |
| Specializes | Sigma-algebra |
| Same as |
Markdown · JSON · Edit on GitHub
What this page needs for W2
- edge instance_of in: at least two examples (0 of 2)
- edge not_instance_of in: at least one non-example (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.
What links here
Definitions
Used by
- Spectral theorem for unbounded self-adjoint operators
- Lebesgue measure
- Capacitability of Borel sets
- Borel determinacy theorem
- Suslin's theorem
- Tightness of measures
- Borel measure
- Haar measure
- Standard Borel space
- Fourier–Stieltjes transform
- Weak convergence of measures
- Borel functional calculus
- Measurable function
- Cylinder set measure
- Borel hierarchy
- Lévy–Prokhorov metric
- De Finetti's theorem
- Spectral theorem for bounded normal operators
- Lebesgue–Stieltjes measure