Euclidean balls have positive finite Lebesgue measure

Depends on

• The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
• Open ball, closed ball and sphere in a metric space
• $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it
• Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed
• Assuming countable choice, every Borel subset of $\mathbb{R}^n$ is Lebesgue measurable
• A box in $\mathbb{R}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure $\prod_{i<n}(b_i-a_i)$, whichever of its faces are included
• 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
• Measures are monotone

Used by

• Bounded mesh ratio gives an h⁻²ˢ exact condition-number bound Corollary
• Poincare-Wirtinger on bounded convex domains by the direct pairwise argument Corollary
• The obstacle reaction is supported on the contact set under measure regularity Corollary
• Uniqueness of classical Dirichlet and compatible Neumann solutions Corollary
• A gradient-only Poincare estimate needs normalisation Counterexample
• Neumann Poisson data require a flux compatibility equation Counterexample
• Poincare-Wirtinger fails on disconnected bounded domains Counterexample
• Strong fractional integration fails at p equal to one Counterexample
• The complementarity product needs extra regularity Counterexample
• The critical Riesz potential can diverge and be essentially unbounded Counterexample
• The critical Sobolev embedding is not compact Counterexample
• The fundamental Hessian is not absolutely locally integrable Counterexample
• The Neumann Poisson problem is not coercive on all of H¹ Counterexample
• A family shrinking nicely to a point Definition
• Density of a measurable set at a point Definition
• Fundamental solution for the positive operator minus Laplacian Definition
• The average of a locally integrable function over a Euclidean ball Definition
• The real Hardy space Hᵖ defined by a radial maximal function Definition
• A Dirac mass has precisely sufficiently negative Sobolev order Example
• Adding an entire harmonic function preserves a Laplace fundamental solution Example
• Dilation determines the Riesz-potential target exponent Example
• Flux normalization on every centered sphere Example
• Heat and Poisson semigroups as Fourier multipliers Example
• Hilbert transform of the line Poisson kernel Example
• Newton shell theorem from harmonic mean values Example
• Piecewise-affine approximation of a measurable coefficient Example
• Sharp Sobolev threshold for a radial power Example
• The heat smoothing time exponent is forced by scaling Example
• The Neumann kernel is spanned by the componentwise constants Example
• A conforming averaging companion for Crouzeix–Raviart functions Lemma
• Ball and cube maximal functions are pointwise comparable Lemma
• Bounded-overlap ball chains in a bounded John domain Lemma
• Cotlar's inequality for maximal truncations Lemma
• Critical images of Euclidean Lipschitz maps are null Lemma
• Dyadic annulus far-field estimates for the maximal function Lemma
• Korn coercivity with a partial Dirichlet segment Lemma
• Lebesgue-point convergence for radial-majorized kernels Lemma
• Level decomposition of an Hᵖ distribution produces atoms Lemma
• Local integrability of the Laplace fundamental kernel Lemma
• Local weak scalar potentials and continuation on a simply connected open set Lemma

…and 29 more results.

Dependency tree · two levels

57 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

• Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Lemma 3.16 (standard reference, not scraped)