A Dirac set function is a probability measure

Depends on

• The Dirac set function at a point
• Probability measures and probability spaces
• Series in the nonnegative extended real line

Used by

• Higher-dimensional simple symmetric walks are transient Corollary
• One-dimensional simple symmetric walk is recurrent Corollary
• Two-dimensional simple symmetric walk is recurrent Corollary
• A submartingale need not have increasing sample paths Counterexample
• An adapted process need not be a martingale Counterexample
• An unbounded predictable transform may lose integrability Counterexample
• Doubling ergodicity depends on the invariant measure Counterexample
• Equal finitely many moments do not determine a law Counterexample
• Taking out an unbounded factor needs integrability Counterexample
• Complex Lp classes and Euclidean test-function conventions Definition
• A boundary atom gives an h1 function without an L1 density Example
• A Dirac probability measure concentrates all mass at one point 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
• Birth–death recurrence through scale products Example
• Borel-Cantelli for the shrinking intervals (0,2⁻ᵏ) under a dyadic atomic measure Example
• Characteristic functions of bernoulli binomial and poisson laws Example
• Chebyshev extremal nodes converge to the arcsine equilibrium measure Example
• Extreme points of the probability measures are Dirac masses Example
• Gambler’s ruin from harmonicity Example
• Green kernel of a biased integer walk Example
• Independent sums via characteristic functions Example
• Integrating against a Dirac measure is evaluation at the point Example
• Point evaluation is represented by a Dirac measure Example
• Square of a martingale minus quadratic compensator Example
• The measure δ₀+λ↾_[0,1] splits into discrete and absolutely continuous parts Example
• The weights 2⁻⁽ᵏ⁺¹⁾ define a probability measure on P(ℕ) Example
• False: constant continuous invariants characterize measure ergodicity False statement
• FALSE: measures are additive on arbitrary countable unions False statement
• Monic polynomial lower bounds for the Chebyshev constant and capacity Lemma
• 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
• Krylov–Bogolyubov existence of an invariant probability Theorem
• Lyapunov drift bound for hitting times Theorem
• Recurrence and transience are class properties Theorem

Cited to discharge well-definedness by The Dirac set function at a point.

Dependency tree · two levels

9 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

• T. Tao, An Introduction to Measure Theory, Example 1.4.29 (standard reference, not scraped)