A continuous map has Borel preimages of Borel sets

Depends on

• The Borel sigma-algebra of a topological space
• Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal
• For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and $f(\overline{A}) \subseteq \overline{f(A)}$

Used by

• Continuous functions on Euclidean spaces are Borel measurable Corollary
• Polar integration may discard the cut locus Corollary
• There is a Lebesgue measurable subset of ℝ that is not Borel Corollary
• Measurable and decomposable operator fields Definition
• The one-dimensional torus and its normalized Haar integral Definition
• The polar surface set function on the unit sphere Definition
• 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
• One-dimensional Dirichlet Green kernel on an interval Example
• Radial second moment of multidimensional Brownian motion Example
• Sharp Sobolev threshold for a radial power Example
• The Cantor function is Borel measurable Example
• The square of the Volterra operator has zero trace Example
• FALSE: every Riemann integrable function on a closed bounded interval is Borel measurable False statement
• A C¹ diffeomorphism maps Lebesgue measurable sets to Lebesgue measurable sets Lemma
• Absolute real powers are Borel measurable and convex Lemma
• Borel change of variables from the compact-support formula and Radon uniqueness Lemma
• Bounded compact data give an everywhere finite Newtonian potential Lemma
• Convex functions have countable supporting line representations Lemma
• Diagonal multipliers form a von Neumann algebra Lemma
• Measurable sections have measurable pointwise inner products Lemma
• Decomposable operators are the commutant of diagonal multiplication Theorem
• Direct integrals of measurable Hilbert fields are Hilbert spaces Theorem
• Far-field asymptotics of compact-source Newtonian potentials Theorem
• Green and harmonic-measure representation with the 2π sign Theorem
• Green representation for classical Poisson data Theorem
• Measurable essentially bounded operator fields act decomposably Theorem
• Newtonian potentials solve the distributional Poisson equation Theorem
• Spectral multiplicity model for separably acting abelian von Neumann algebras Theorem
• The Borel product of Rᵐ and Rⁿ is the Borel sigma-algebra of Rᵐ⁺ⁿ Theorem

Dependency tree · two levels

10 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, Remark 1.4.15 (standard reference, not scraped)