Vol. 40, No. 3, 1972

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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

Suppose that {ℳk} is an increasing sequence of sub σlattices of a σ-algebra 𝒜 of subsets of a non-empty set Ω. Let be the sub σ-lattice generated by kk. Suppose that LΦ is an associated Orlicz space of 𝒜-measurable functions, where Φ satisfies the Δ2-condition, and let h LΦ. It is verified that the Radon-Nikodym derivative, fk, of h given ℳ′ is in LΦ and shown that the sequence {fk} converges to f in LΦ, where f is the Radon-Nikodym derivative of h given

Mathematical Subject Classification
Primary: 60G45
Secondary: 28A45
Received: 7 April 1971
Published: 1 March 1972
Richard Brian Darst
Gene Allen DeBoth