Assuming Dependent Choice, uniqueness of the RMK representing measure among Radon measures

Depends on

• The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain
• Positive functionals on C_c(X) are integration against a Radon measure
• Radon measure on an LCH space
• LCH Urysohn cutoff

Used by

• Existence of left and right Haar measures Corollary
• Uniqueness of finite Borel measures from their Fourier transforms Corollary
• Left, right, and bi-invariant Borel measures Definition
• Haar measure on an infinite product of compact groups Example
• An RMK functional determines every Borel representing measure uniquely False statement
• Compact-open neighbourhoods on the dual give a neighbourhood basis on the group Lemma
• Continuous functional calculus produces a regular PVM Lemma
• Continuous quotient translation cocycle Lemma
• Density of averaged covariant generators Lemma
• Fourier-Stieltjes transforms determine finite Radon measures Lemma
• Haar change of variables under inversion Lemma
• Haar measure on an abelian group is invariant under inversion Lemma
• Haar null classes and Borel descent on a homogeneous space 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 C₀(X) functionals have finite regular representing measures Lemma
• Relative property (T) for SL2(R) semidirect R2 Lemma
• Sobolev pullback and Green identity on Lipschitz graphs Lemma
• Criterion for an invariant quotient measure Proposition
• Compatible dual Haar normalisation Theorem
• Existence of rho-functions and quotient measure classes Theorem
• Independence of rho and equivalent quotient representative Theorem
• The distributional Riesz functional of a subharmonic function is a positive Radon measure Theorem
• The principle of descent and the logarithmic domination principle 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., Chapter 7 (standard reference, not scraped)