Dynkin's pi-lambda theorem

Depends on

• The lambda-system generated by a pi-system is closed under finite intersections
• A lambda-system closed under finite intersections is a sigma-algebra
• The generated lambda-system exists and is minimal
• Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal

Used by

• A C¹ diffeomorphism satisfies the change-of-variables formula for L¹ functions Corollary
• Standard borel spaces have countable generating and measure determining algebras Corollary
• Weak limits are unique Corollary
• Gaussian AR(1) chain Example
• Radial second moment of multidimensional Brownian motion Example
• Successive Brownian exit segments are independent copies Example
• Conditional-independence equivalences and preservation Lemma
• Conditional-independence splice lemma Lemma
• Conditioning a known state and independent noise Lemma
• Finite measures agreeing on a generating pi-system and on the whole space are equal Lemma
• Levy prokhorov distance is a metric Lemma
• Rational conditional distribution functions produce real regular kernels Lemma
• Regular conditional kernels factor through a standard borel conditioning variable Lemma
• Assuming countable and dependent choice, countable products of arbitrary probability spaces Theorem
• Assuming countable choice, Borel probability measures on Polish spaces are inner regular Theorem
• Assuming the Axiom of Choice, Kolmogorov extension for arbitrary families of standard Borel coordinate spaces Theorem
• Borel harmonicity and comparison of harmonic measure Theorem
• Brownian reflection principle Theorem
• Brownian time inversion Theorem
• Brownian-filtration martingale representation Theorem
• Density of elementary predictable processes in predictable L2 Theorem
• Direct integrals of measurable Hilbert fields are Hilbert spaces Theorem
• Disintegration of a joint law on standard borel spaces Theorem
• Existence of regular conditional distributions for standard borel targets Theorem
• Finite-dimensional distributions determine a process law on the cylinder sigma-algebra Theorem
• Finite-dimensional laws of a Markov chain Theorem
• Future-path Markov property Theorem
• Independent pi-systems generate independent sigma-algebras Theorem
• Ionescu-Tulcea construction of a Markov chain Theorem
• Markov property for bounded future path functionals Theorem
• Measurability of integration against a kernel Theorem
• Portmanteau theorem Theorem
• Rapid filters are not Lebesgue measurable Theorem
• Strong Markov property of Brownian motion Theorem
• Uniqueness of Wiener measure Theorem

Dependency tree · two levels

10 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

• A. Dembo, Probability Theory lecture notes, Theorem 1.1.38 (standard reference, not scraped)