Measure convergent sequences in Lebesgue spaces and Fatou's lemma

Abstract

Let ( f n ) be a sequence of positive P -integrable functions such that (∫ f n dP ) n converges. We prove that ( f n ) converges in measure to if and only if equality holds in the generalised Fatou's lemma. Let f ∞ be an integrable function such that (∥ f n − f ∞ ∥ 1 ) n converges. We present in terms of the modulus of uniform integrability of ( f n ) necessary and sufficient conditions for ( f n ) to converge in measure to f ∞ .

Locations

• Bulletin of the Australian Mathematical Society - View - PDF