a1 Université de Reims, Département de Mathématiques, Moulin de la Housse, B P 347, 51062 Reims Cedex, France e-mail: [email protected]
Let (fn) be a sequence of positive P-integrable functions such that (∫ fndP)n converges. We prove that (fn) converges in measure to if and only if equality holds in the generalised Fatou's lemma. Let f∞ be an integrable function such that (fn − f∞1)n converges. We present in terms of the modulus of uniform integrability of (fn) necessary and sufficient conditions for (fn) to converge in measure to f∞.
(Received October 16 1995)