Bulletin of the Australian Mathematical Society

Research Article

Measure convergent sequences in Lebesgue spaces and Fatou's lemma

Heinz-Albrecht Kleia1

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 S0004972700017652_inline1 if and only if equality holds in the generalised Fatou's lemma. Let f be an integrable function such that (xs2225fnfxs22251)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)