The Lebesgue decomposition, Radon-Nikodym derivative, conditional expectation, and martingale convergence for lattices of sets

Richard Brian Darst

Vol. 35 (1970), No. 3, 581–600

DOI: 10.2140/pjm.1970.35.581

Abstract

In the setting of additive set functions defined on lattices of sets, a Lebesgue decomposition and a Radon-Nikodym derivative are constructed and characlerized. In the appropriate case (L₂), the Radon-Nikodym derivative is shown to be the conditional expectation. Finally, a martingale convergence theorem for Radon-Nikodym derivatives is obtained.

Mathematical Subject Classification 2000

Primary: 28A10

Secondary: 60B99

Milestones

Received: 24 July 1969

Revised: 10 April 1970

Published: 1 December 1970