LCH Urysohn cutoff

Depends on

• Compact support, $C_c(X)$, and $C_0(X)$
• In a locally compact Hausdorff space every open set containing a point contains an open set containing it whose closure is compact and still inside; such a space is regular
• Urysohn's lemma, under the axiom of dependent choice: in a normal space two disjoint closed sets are separated by a continuous function into $[0,1]$, and conversely such a space is normal

Used by

• Naive inversion is not the L1 involution on a nonunimodular group Counterexample
• Every positive linear functional on C_c(X) is uniformly sup-norm bounded False statement
• A finite compactly supported partition of unity near a compact set Lemma
• A system of imprimitivity integrates to a nondegenerate representation of the transformation algebra Lemma
• Bruhat cutoff normalized along H-fibers Lemma
• Characters of the L1 algebra of an abelian group Lemma
• Compact lifts and averaging onto C_c(G/H) Lemma
• Compact-open neighbourhoods on the dual give a neighbourhood basis on the group Lemma
• Compact-set formula and local finiteness of the RMK measure Lemma
• Compactly supported kernels admit commuting radon integrals Lemma
• Compactly supported nonnegative transform bumps on the dual Lemma
• Continuous characters separate points of an LCA group Lemma
• Density of averaged covariant generators Lemma
• Haar candidate sets have the finite intersection property Lemma
• Haar covering functionals are asymptotically additive Lemma
• Haar integral comparison inequality Lemma
• Haar measure on an abelian group is invariant under inversion Lemma
• Haar regularization of transitive unitary cocycles Lemma
• LCA Fourier transforms form a dense algebra in C0 of the dual Lemma
• Nonzero multiplicative functionals on L¹ of an LCA group are Fourier evaluations Lemma
• Positive C₀(X) functionals have finite regular representing measures Lemma
• Spectral measure of a unitary representation of an abelian group, covariance, and ergodicity Lemma
• The Bochner functional extends and has a Radon representing measure Lemma
• The L1 action of a strongly continuous unitary representation Lemma
• The RMK functional outer content is well defined Lemma
• Translation continuity and normalised local approximate identities on an LCA group Lemma
• The L1 group algebra has a unit exactly when the group is discrete Proposition
• Assuming Dependent Choice, uniqueness of the RMK representing measure among Radon measures Theorem
• Bochner's theorem for LCA groups Theorem
• C_c(X) is dense in Lᵖ(mu) for a Radon measure Theorem
• Compatible dual Haar normalisation Theorem
• Every commutative C star algebra has an approximate unit Theorem
• L1 group algebras have a contractively bounded approximate identity Theorem
• Locally compact Gelfand duality Theorem
• Open sets are Caratheodory measurable for the RMK outer measure Theorem
• The bounded complex dual of C₀(X) is regular complex measures Theorem
• The modular function is a continuous homomorphism Theorem
• The regular representations are unitary, strongly continuous, and the left one is faithful Theorem
• Uniqueness of left Haar measure up to scale Theorem
• Weil formula with a rho-function Theorem

Dependency tree · two levels

28 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., Proposition 7.1.9 (standard reference, not scraped)