The Borel sigma-algebra of a topological space

Depends on

• The sigma-algebra generated by a family of sets
• Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal
• Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison

Used by

• L(ℝⁿ) is exactly the completion of the restriction of λₙ to the Borel sets Corollary
• Polar integration may discard the cut locus Corollary
• A non-sigma-finite premeasure has distinct Borel extensions Counterexample
• Neumann Poisson data require a flux compatibility equation Counterexample
• The fundamental Hessian is not absolutely locally integrable Counterexample
• A Borel measure on ℝ that is finite on compact sets Definition
• An atom of a measure on ℝ Definition
• Borel master codes for null and meagre sets Definition
• Borel measurable and Lebesgue measurable functions on ℝⁿ Definition
• Borel probability law on a polish space Definition
• Extended-real-valued measurable functions Definition
• Harmonic measure on a bounded regular plane domain Definition
• Pointwise Borel nonnegative densities Definition
• Radon measure on an LCH space Definition
• Random elements and real random variables Definition
• The Borel sigma-algebra on the extended real line Definition
• The one-dimensional torus and its normalized Haar integral Definition
• The polar surface set function on the unit sphere Definition
• The sigma-algebra generated by a function Definition
• Weak convergence of borel probability measures Definition
• Wiener measure on continuous path space Definition
• Assuming choice, the completion of the Borel Dirac measure at zero is defined on every subset of the real line Example
• F-sigma and G-delta subsets of the real line are Borel Example
• Harmonic measure of a real interval from the upper half-plane Example
• Harmonic measure of an arc of the unit circle Example
• Matching C¹ pieces across a hyperplane have no jump derivative Example
• Newton shell theorem from harmonic mean values Example
• Newtonian potential of radial compact data Example
• Sharp Sobolev threshold for a radial power Example
• The rationals are Borel and F-sigma but neither open nor closed nor G-delta Example
• The square of the Volterra operator has zero trace Example
• Assuming the Axiom of Choice, every Lebesgue measurable subset of ℝ is a Borel set False statement
• Every translation-invariant measure on the Borel sets of ℝ is a nonnegative multiple of Lebesgue measure False statement
• FALSE: the extension of a premeasure is always unique False statement
• A shear sends the unit cube to a set of Lebesgue measure one Lemma
• A translation-invariant Borel measure giving the unit cube measure one gives each generation-k dyadic cube measure 2⁻ᵏⁿ Lemma
• Borel sigma-algebra of continuous path space is generated by coordinates Lemma
• Counting measure on a discrete group is Haar, Haar measures there are its multiples, and integrals against them are sums Lemma
• Dyadic coding supplies coin measure and its completed Lebesgue transfer Lemma
• Elementary bounds on ideal cardinal invariants Lemma

…and 25 more results.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• T. Tao, An Introduction to Measure Theory, Definition 1.4.16 (standard reference, not scraped)
• R. F. Bass, Real Analysis for Graduate Students, version 5.0, Section 2.1 (standard reference, not scraped)