Vol. 30, No. 2, 1969

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Martingales of vector valued set functions

J. Jerry Uhl, Jr.

Vol. 30 (1969), No. 2, 533–548

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 LΦ of measurable point functions. First, a conditionaI expectation PB, operating on finitely additive set functions, is introduced and related to the usual conditional expectation EB operating on L1 by the equality

(∗)P (F)(E) =   EB (f)dμE ∈ Σ
B          E

where ,Σ) is a measure space, B is a sub σ-field of Σ and F(E) = Efdμ for E Σ.

Then, with the use of PB martingales of set functions are defined and their convergence in appropriate V Φ spaces is investigated. In addition, in the countably additive case, the results obtained for martingales of set functions are related to martingales of measurable point functions and extensions of certain results of Scalora, Chatterji, and Helms are obtained.

Mathematical Subject Classification
Primary: 46.35
Secondary: 28.00
Received: 22 February 1968
Published: 1 August 1969
J. Jerry Uhl, Jr.