Talagrand's proof of the sufficiency of existence of a majorizing measure for the sample boundedness of processes with bounded increments used a contraction from a certain ultrametric space. We give a short proof of existence of such an ultrametric using admissible sequences of nets.
The paper contains some sufficient conditions for Marczewski-Burstin representability of an algebra 𝓐 of sets which is isomorphic to 𝓟(X) for some X. We characterize those algebras of sets which are inner MB-representable and isomorphic to a power set. We consider connections between inner MB-representability and hull property of an algebra isomorphic to 𝓟 (X) and completeness of an associated quotient algebra. An example of an infinite universally MB-representable algebra is given.
ℒ denotes the Lebesgue measurable subsets of ℝ and denotes the sets of Lebesgue measure 0. In 1914 Burstin showed that a set M ⊆ ℝ belongs to ℒ if and only if every perfect P ∈ ℒ$ℒ0 which is a subset of or misses M (a similar statement omitting “is a subset of or” characterizes ). In 1935, Marczewski used similar language to define the σ-algebra (s) which we now call the “Marczewski measurable sets” and the σ-ideal which we call the “Marczewski null sets”. M ∈ (s) if every perfect set P has...
We consider the family 𝓜 of measures with values in a reflexive Banach space. In 𝓜 we introduce the notion of a Markov operator and using an extension of the Fortet-Mourier norm we show some criteria of the asymptotic stability. Asymptotically stable Markov operators can be used to construct coloured fractals.