Vol. 87, No. 2, 1980

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The Radon-Nikodým property, σ-dentability and martingales in locally convex spaces

Leo Egghe

Vol. 87 (1980), No. 2, 313–322

In this paper we give relations between the Radon-Nikodym-Property (RNP), in sequentially complete locally convex spaces, mean convergence of martingales, and σ-dentability. (RNP) is equivalent with the property that a certain class of martingales is mean convergent, while σ-dentability is equivalent with the property that the same class of martingales is mean Cauchy. We give an example of a σ-dentable space not having the (RNP). It is also an example of a sequentially incomplete space of integrable functions, the range space being sequentially complete.

Mathematical Subject Classification 2000
Primary: 46E40
Secondary: 46A99
Received: 2 November 1978
Revised: 8 March 1979
Published: 1 April 1980
Leo Egghe