Martingales of strongly measurable Pettis integrable functions

Uhl, J. J.

doi:10.1090/S0002-9947-1972-0293708-X

Martingales of strongly measurable Pettis integrable functions
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Trans. Amer. Math. Soc. 167 (1972), 369-378 Request permission

Erratum: Trans. Amer. Math. Soc. 181 (1973), 507.

Abstract:

This paper deals with convergence theorems for martingales of strongly measurable Pettis integrable functions. First, a characterization of those martingales which converge in the Pettis norm is obtained. Then it is shown that a martingale which is convergent in the Pettis norm converges to its limit strongly in measure and, if the index set is the positive integers, it converges strongly almost everywhere to its limit. The second part of the paper deals with the strong measure and strong almost everywhere convergence of martingales which are not necessarily convergent in the Pettis norm. The resulting theorems here show that ${L^1}$-boundedness can be considerably relaxed to a weaker control condition on the martingale by the use of some facts on finitely additive vector measures.

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