Martingales of vector valued set functions
J. J. Uhl · Pacific Journal of Mathematics · 1969
This paper is concerned with the norm convergence of Banach space valued martingales in Orlicz spaces whose underlying measure is (possibly) only finitely additive.Because of the possible incompleteness of these Orlicz spaces of measurable point functions, this subject will be treated in the setting of Orlicz spaces of set functions V φ rather than the corresponding spaces L φ of measurable point functions.First, a conditional expectation P B , operating on finitely additive set functions, is introduced and related to the usual conditional expectation E B operating on L 1 by the equality ( * ) PB(F)(E) = ( E\f)dμ Ee Σ where (Ω 9 Σ, μ) is a measure space, B is a sub 0, there exists a o > 0 such that μ(E) < δ implies ||-FCE7)|| < ε.