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    Giner, Emmanuel 2012. Fatou’s lemma and lower epi-limits of integral functionals. Journal of Mathematical Analysis and Applications, Vol. 394, Issue. 1, p. 13.


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  • Bulletin of the Australian Mathematical Society, Volume 54, Issue 2
  • October 1996, pp. 197-202

Measure convergent sequences in Lebesgue spaces and Fatou's lemma

  • Heinz-Albrecht Klei (a1)
  • DOI: http://dx.doi.org/10.1017/S0004972700017652
  • Published online: 01 April 2009
Abstract

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 (∥fnf1)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.

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[1]E.J. Balder , ‘On equivalence of strong and weak convergence in L1-spaces under extreme point conditions’, Israel J. Math. 75 (1991), 2147.

[2]H. Brezis and E. Lieb , ‘A relation between pointwise convergence of functions and convergence of functional’, Proc. Amer. Math. Soc. 88 (1983), 486490.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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