Orlicz space convergence of martingales of Radon-Nikodým derivatives given a σ-lattice

David Alan Legg

Vol. 67 (1976), No. 1, 209–214

DOI: 10.2140/pjm.1976.67.209

Abstract

Let {M_k} be an increasing sequence of sub σ-lattices of a σ-algebra 𝒜, and let M be the σ-lattice generated by ⋃ _kM_k. Let LΦ be an associated Orlicz space of 𝒜-measurable functions, where Φ does not necessarily satisfy the Δ₂-condition. Given h ∈ LΦ, let f_k be the Radon-Nikodym derivative of h given M_k. Necessary and sufficient conditions are given on h to insure that {f_k} converges in LΦ to f, where f is the Radon-Nikodym derivative of h given M. The situation where f is valued in a Banach space with basis is also examined.

Mathematical Subject Classification

Primary: 60G45, 60G45

Milestones

Received: 5 May 1976

Published: 1 November 1976