Chapter 13. On the weak convergence in $\mathbb{L}^p$ Spaces

Abstract

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 $\mathbb{L}^{p}$.