Norm convergence of martingales of Radon-Nikodym derivatives given a σ-lattice

Richard Brian Darst and Gene Allen DeBoth

Vol. 40 (1972), No. 3, 547–552

DOI: 10.2140/pjm.1972.40.547

Abstract

Suppose that {ℳ_k} is an increasing sequence of sub σ− lattices of a σ-algebra 𝒜 of subsets of a non-empty set Ω. Let ℳ be the sub σ-lattice generated by ⋃ _kℳ_k. Suppose that L^Φ is an associated Orlicz space of 𝒜-measurable functions, where Φ satisfies the Δ₂-condition, and let h ∈ L^Φ. It is verified that the Radon-Nikodym derivative, f_k, of h given ℳ′ is in L^Φ and shown that the sequence {f_k} converges to f in L^Φ, where f is the Radon-Nikodym derivative of h given ℳ

Mathematical Subject Classification

Primary: 60G45

Secondary: 28A45

Milestones

Received: 7 April 1971

Published: 1 March 1972