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Locally finite Borel measures on second-countable LCH spaces are regular
Statement
Assume the Axiom of Countable Choice . Every Borel measure finite on compact sets on a second-countable LCH space is regular.
Facts & Assumptions
Given: Countable Choice (The Axiom of Countable Choice ()), is second-countable and LCH, and is finite on compact sets.
An LCH space has a base of open sets with compact closure. (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)
Under Countable Choice, if every open set is sigma-compact, compact-finite Borel measures are regular (Sigma-compact open sets make locally finite Borel measures regular). The stated choice assumption is spent only in this theorem.
Proof
Let be a given countable base and let be open. Take all indices for which is compact and contained in ; this is a specified subfamily of the given base and needs no selection. For each , [L1] gives an open with and compact. Choose one with ; its closure is a closed subset of the compact , hence compact and contained in . Thus the selected subfamily covers , and is the union of the corresponding countably many compact closures.
Thus every open set is sigma-compact. Apply [L2] under the stated Countable Choice premise to obtain regularity of .
Depends on
- The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
- Sigma-compact open sets make locally finite Borel measures regular
- Second countability: an at most countable basis for the topology
- 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
Used by
- Uniqueness of finite Borel measures from their Fourier transforms Corollary
- The Poisson integral of a finite complex boundary measure Definition
- A boundary atom gives an h1 function without an L1 density Example
- A locally integrable density functional is represented by g dlambda Example
- Little groups for the real ax+b group and its orientation-preserving subgroup Example
- Multiplication operators: domain, spectral measure and spectrum Example
- Multiplicity-two diagonal representation Example
- Pvm of a multiplication operator Example
- Riesz measure of a log modulus records the holomorphic zeros Example
- The modular function of the affine group of the line Example
- Borel change of variables from the compact-support formula and Radon uniqueness Lemma
- Complex circle measures have finite regular total variation under countable choice Lemma
- Haar null classes and Borel descent on a homogeneous space Lemma
- Multiplicity model of a projection-valued measure over a standard Borel base Lemma
- Operators commuting with a generating family of multiplications Lemma
- Relative property (T) for SL2(R) semidirect R2 Lemma
- Signed narrow convergence controls variation tails and Bochner moments Lemma
- Singular circle measures have Poisson integral tending nontangentially to zero almost everywhere Lemma
- Spectral alias identity for fixed regular-grid sampling Lemma
- Spherical harmonics form a complete basis with the addition formula Lemma
- Strict positivity of logarithmic energy for a zero-mass signed charge Lemma
- The circle maximal function is weak type one one for finite measures Lemma
- Green and harmonic-measure representation with the 2π sign Theorem
- h1 is isometric to finite regular complex boundary measures Theorem
- Herglotz spectral measure and the summable-covariance density Theorem
- hᵖ is the Poisson image of Lp for 1<p<=infinity Theorem
- Local Riesz decomposition of a plane subharmonic function Theorem
- Positive smooth densities give Radon volume Theorem
- Properties of the singular functions S_μ Theorem
Dependency tree · two levels
27 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)