Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable

Depends on

• Assuming countable choice, $\mathcal{L}(\mathbb{R}^n)$ is a sigma-algebra containing every elementary set and $\lambda_n$ is a complete measure extending elementary volume
• The sigma-algebra generated by the half-open boxes of $\mathbb{R}^n$ is the Borel sigma-algebra
• The Borel sigma-algebra of a topological space
• Sigma-algebras
• 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
• Lebesgue measurable sets, the family $\mathcal{L}(\mathbb{R}^n)$, and the restricted set function $\lambda_n$
• Elementary sets: the finite unions of half-open boxes in $\mathbb{R}^n$
• The Axiom of Countable Choice ($\mathrm{AC}_\omega$)

Used by

• Every subset of ℝⁿ has a G_δ measurable hull of the same outer measure Corollary
• L(ℝⁿ) is exactly the completion of the restriction of λₙ to the Borel sets Corollary
• There is a Lebesgue measurable subset of ℝ that is not Borel Corollary
• A nonintegrable observable with divergent ergodic averages Counterexample
• A null set can fail to be the discontinuity set of any function Counterexample
• Neumann Poisson data require a flux compatibility equation Counterexample
• Not every compact set is conformally removable Counterexample
• Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
• Pointwise modification can destroy path continuity Counterexample
• The critical Riesz potential can diverge and be essentially unbounded Counterexample
• The indicator of a fat Cantor set is upper semicontinuous and equal almost everywhere to no Riemann integrable function Counterexample
• Lebesgue inner measure on the real line Definition
• The Bergman space A²(Ω) and the Bergman kernel Definition
• The one-dimensional torus and its normalized Haar integral Definition
• The polar surface set function on the unit sphere Definition
• A clipped affine function keeps its zero region Example
• An open dense set of measure less than 1 is the monotone L¹-limit of Riemann integrable indicators, but its indicator is not Riemann integrable Example
• Matching C¹ pieces across a hyperplane have no jump derivative Example
• Newtonian potential of radial compact data Example
• Sharp Sobolev threshold for a radial power Example
• The absolute value has a weak first derivative Example
• The positive-type Gaussian on the real line and its cyclic model Example
• Birkhoff's theorem requires integrability False statement
• A measurable set of positive finite measure occupies more than any prescribed proportion of some dyadic cube Lemma
• A separated characteristic disk has a minimal nonidentity simple cycle Lemma
• A shear sends the unit cube to a set of Lebesgue measure one Lemma
• A subset of ℝⁿ with open supersets of arbitrarily small excess is Lebesgue measurable Lemma
• Agreement of Borel overlap integrals Lemma
• An area-minimal three-sector homoclinic cycle has identity inward holonomy Lemma
• Bounded compact data give an everywhere finite Newtonian potential Lemma
• Compact sets of positive area are not conformally removable Lemma
• Compact subsets of lines and round circles are removable for quasiconformal maps Lemma
• Dyadic coding supplies coin measure and its completed Lebesgue transfer Lemma
• Euclidean balls have positive finite Lebesgue measure Lemma
• For a Lebesgue measurable set and every positive ε there is an open superset whose difference from it has outer measure below ε Lemma
• Fourier uniqueness for continuous functions on the Euclidean torus Lemma
• Lebesgue-point convergence for radial-majorized kernels Lemma
• Local integrability of the Laplace fundamental kernel Lemma
• Monomials form complete orthogonal systems of the Bergman spaces of the disc, the ball and the polydisc Lemma
• Near and far bounds for a Riesz potential Lemma

…and 19 more results.

Dependency tree · two levels

43 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

• John K. Hunter, Measure Theory (UC Davis lecture notes), Proposition 2.21 (standard reference, not scraped)
• E. A. Carlen, Notes on Lebesgue Measure on $\mathbb{R}^n$ and $S^{n-1}$ (Rutgers Math 501), Proposition 1.4 (standard reference, not scraped)