C_c(X) is dense in L^p(mu) for a Radon measure

Depends on

• The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain
• Radon measure on an LCH space
• Compact support, $C_c(X)$, and $C_0(X)$
• LCH Urysohn cutoff
• Simple functions with finite-measure support are dense in $L^p(\mu)$ for $1 \le p < \infty$
• The space $L^p(\mu)$ as the quotient by null functions

Used by

• Bounded Sobolev sequences have strongly convergent subsequences with the weak limit as limit Corollary
• Hilbert and Riesz transforms are bounded on weighted L-p Corollary
• Left and right regular representations on L2(G) Definition
• Multiplicity-two diagonal representation Example
• C_c(X) is dense in Lⁱnfinity(mu) for every Radon measure False statement
• Central continuous approximate identities Lemma
• Compact-open neighbourhoods on the dual give a neighbourhood basis on the group Lemma
• Compactly supported nonnegative transform bumps on the dual Lemma
• Completeness of the complex Haar L1 and L2 spaces and density of Cc Lemma
• Density of averaged covariant generators Lemma
• Differentiation of L-one functions for a doubling weight Lemma
• L¹ of an LCA group is a commutative Banach star algebra under convolution Lemma
• Nonzero multiplicative functionals on L¹ of an LCA group are Fourier evaluations Lemma
• Positive convolution squares form a dense inversion core Lemma
• Positive definite functions give positive bounded functionals on the transform core Lemma
• Quadratic path energy is lower semicontinuous Lemma
• Scalar unitisation of L¹ of an LCA group: characters, spectrum and identity criterion Lemma
• Smooth compactly supported tests are dense for finite Radon L2 measures Lemma
• Spherical harmonics form a complete basis with the addition formula Lemma
• The character topology on L¹ of an LCA group is the compact-open topology Lemma
• The Plancherel transform range is dense in L² of the dual Lemma
• Translation continuity and normalised local approximate identities on an LCA group Lemma
• Unitary intertwiners preserve direct-integral fiber dimension Lemma
• Cyclic spectral representation Theorem
• Fourier inversion for integrable transforms on LCA groups Theorem
• Plancherel isometric extension on LCA groups Theorem
• Pontryagin biduality: the evaluation map is a topological isomorphism Theorem
• Riemann-Lebesgue lemma on LCA groups Theorem
• Spectral multiplicity model for separably acting abelian von Neumann algebras Theorem
• Superposition for finite-action Euclidean continuity equations Theorem
• Weighted L-p bounds for standard Calderon-Zygmund maximal truncations Theorem

Dependency tree · two levels

30 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

• Donald L. Cohn, Measure Theory, 2nd ed., Chapter 7 (standard reference, not scraped)