The Dirac set function at a point

Depends on

• Sigma-algebras

Used by

• Two-dimensional simple symmetric walk is recurrent Corollary
• Taking out an unbounded factor needs integrability Counterexample
• Total variation can exceed the absolute value of the set value Counterexample
• Complex Lp classes and Euclidean test-function conventions Definition
• Robin constant and logarithmic capacity of a compact set Definition
• Standard normal and normal laws Definition
• A boundary atom gives an h1 function without an L1 density Example
• A singular inner function generated by a point mass Example
• A step function generates a finite atomic measure Example
• Assuming choice, the completion of the Borel Dirac measure at zero is defined on every subset of the real line Example
• Chebyshev extremal nodes converge to the arcsine equilibrium measure Example
• Finite and countable planar sets have zero logarithmic capacity Example
• Integrating against a Dirac measure is evaluation at the point Example
• Point evaluation is represented by a Dirac measure Example
• Riesz measure of a log modulus records the holomorphic zeros Example
• The measure δ₀+λ↾_[0,1] splits into discrete and absolutely continuous parts Example
• The signed measure delta₁ minus delta_-1 has the obvious Hahn and Jordan decomposition Example
• The weights 2⁻⁽ᵏ⁺¹⁾ define a probability measure on P(ℕ) Example
• FALSE: measures are additive on arbitrary countable unions False statement
• FALSE: total variation always equals the absolute value of the set value False statement
• Monic polynomial lower bounds for the Chebyshev constant and capacity Lemma
• The Dieudonne measure and top-point Dirac mass agree on continuous functions Lemma
• A Dirac set function is a probability measure Proposition
• A measure on a finite sigma-algebra is a finite weighted sum over its atoms Theorem
• Every finite Borel measure on ℝ splits as an atomic part plus an atomless part Theorem
• Every measure on a countable discrete space is its weighted sum of Dirac measures Theorem
• Fekete–Szegő equality of logarithmic capacity, transfinite diameter, and Chebyshev constant Theorem
• h1 is isometric to finite regular complex boundary measures Theorem

Dependency tree · two levels

2 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

• S. Axler, Measure, Integration & Real Analysis, Example 2.55 (standard reference, not scraped)