Alphabeta MathHow statement and proof provenance work
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Regular complex Borel measures
Definition
Let be a locally compact Hausdorff space. A complex Borel measure on is regular if its total variation is a regular Borel measure in the preceding, all-Borel sense. This total-variation convention is the one that controls and the norm in the representation theorem.
Depends on
Used by
- The circle maximal function and nontangential approach regions Definition
- The Poisson integral of a finite complex boundary measure Definition
- A boundary atom gives an h1 function without an L1 density Example
- A character is positive definite Example
- Continuous functional calculus produces a regular PVM Lemma
- Fourier-Stieltjes transforms determine finite Radon measures Lemma
- The Bochner functional extends and has a Radon representing measure Lemma
- The Plancherel transform range is dense in L² of the dual Lemma
- Unitary intertwiners preserve direct-integral fiber dimension Lemma
- Bochner's theorem for LCA groups Theorem
- Pontryagin biduality: the evaluation map is a topological isomorphism Theorem
- Positive harmonic boundary measures and compact normalized families Theorem
- The bounded complex dual of C₀(X) is regular complex measures Theorem
Dependency tree · two levels
8 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 7.3.4 (standard reference, not scraped)