Open Access
VOL. 1 | 2018 Chapter 13. On the weak convergence in $\mathbb{L}^p$ Spaces

Editor(s) Hamet SEYDI, Gane Samb LO, Aboubakary DIAKHABY


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

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

  • The function $\={f}$ belongs to $\mathcal{L}^{p}$ and $(\={f})_{n \ge 0}$ weakly converges to $\={f}$ in $\mathbb{L}^{p}$.

  • The sequence $(\={f})_{n \ge 0}$ weakly converges in $\mathbb{L}^{p}$.

  • The sequence is $(\={f})_{n \ge 0}$ is bounded $\mathbb{L}^{p}$.


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

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

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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