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Measure convergent sequences in Lebesgue spaces and Fatou's lemma

Published online by Cambridge University Press:  17 April 2009

Heinz-Albrecht Klei
Heinz-Albrecht Klei
Affiliation:
Université de Reims, Département de Mathématiques, Moulin de la Housse, B P 347, 51062 Reims Cedex, France e-mail: [email protected]
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Abstract

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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∞.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 54 , Issue 2 , October 1996 , pp. 197 - 202
Copyright
Copyright © Australian Mathematical Society 1996

References

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