Probability measures and probability spaces

Depends on

• Measures on sigma-algebras
• Measure spaces

Used by

• Lyapunov's moment inequality on a probability space Corollary
• Normalised positive definite functions correspond to probability measures Corollary
• Identical distribution without independence can defeat the mean law Counterexample
• Pointwise modification can destroy path continuity Counterexample
• Taking out an unbounded factor needs integrability Counterexample
• Two four-point probability measures agree on a generating family that is not a pi-system Counterexample
• A consistent family of finite-dimensional distributions Definition
• A metric for convergence in probability Definition
• Almost-sure convergence of real random variables Definition
• Borel probability law on a polish space Definition
• Convergence in probability Definition
• Filtration and filtered probability space Definition
• Independent families of event classes Definition
• Invariant and stationary distribution for a Markov kernel Definition
• Law or distribution of a random element Definition
• Levy prokhorov metric Definition
• Random elements and real random variables Definition
• Robin constant and logarithmic capacity of a compact set Definition
• Standard normal and normal laws Definition
• Tight family of probability measures Definition
• Total variation distance for probability laws Definition
• Weak convergence of borel probability measures Definition
• A character is positive definite Example
• A countable probability space with geometric weights Example
• Almost sure frequency of heads Example
• Arcsine equilibrium measure and capacity of a segment Example
• Bertrand's chord paradox is a model-specification problem Example
• Capacity of a disc and its circular equilibrium measure Example
• Finite and countable planar sets have zero logarithmic capacity Example
• Negative drift gives a finite mean small-set hit Example
• Nonidentical strong law under summable normalized variances Example
• The uniform random variable on [0,1] Example
• The weights 2⁻⁽ᵏ⁺¹⁾ define a probability measure on P(ℕ) Example
• FALSE: agreement on an arbitrary generating family determines a measure False statement
• Basic identities for a probability measure Lemma
• Compact capacity-zero sets and subharmonic minus-infinity loci Lemma
• Monic polynomial lower bounds for the Chebyshev constant and capacity Lemma
• The law of a random element is a probability measure Lemma
• A Dirac set function is a probability measure Proposition
• Reciprocity inequality for logarithmic potentials Proposition

…and 9 more results.

Dependency tree · two levels

5 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, Definition 12.1 (standard reference, not scraped)