Open Access
VOL. 1 | 2018 Chapter 13. On the weak convergence in Lp Spaces
Chapter Author(s) Hamet SEYDI
Editor(s) Hamet SEYDI, Gane Samb LO, Aboubakary DIAKHABY

Abstract

The aim of this paper is to prove the following theorem.

Theorem 34. Let X be a locally Hausdorff compact space, μ a Radon Nykodym on X and (fn) be a sequence of measurable functions (with respect to μ) belonging to Lp(X,μ) which converges in measure to a measurable function. Let \=g stand for the equivalence class of a measurable function g with the equivalence relation R induced by the v-a.e equality and Lp(X,μ) be the quotient by R. Then the following conditions are equivalent.

  • The function \=f belongs to Lp and (\=f)n0 weakly converges to \=f in Lp.

  • The sequence (\=f)n0 weakly converges in Lp.

  • The sequence is (\=f)n0 is bounded Lp.

Information

Published: 1 January 2018
First available in Project Euclid: 26 September 2019

Digital Object Identifier: 10.16929/sbs/2018.100-03-01

Subjects:
Primary: 28A25 , 28A51 , 28C05

Keywords: integration theory , locally compact space , radon measures , weak convergence

Rights: Copyright © 2018 The Statistics and Probability African Society

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