Let $(\Omega, \mathcal F, \mu)$ be a $\sigma$-finite measure space. I'm trying to solve below exercise in Brezis' Functional Analysis, i.e.,
Let $p \in (1, \infty)$ and $p'$ its Hölder conjugate. Let $f_n,f,g:\Omega \to \mathbb R$ be measurable functions such that $\sup_n \|f_n\|_p < \infty$ and that $\| f_n - f \|_1 \to 0$. Then $f_n \to f$ in the weak topology $\sigma(L^p, L^{p'})$.
Could you confirm if my attempt is fine?
Proof Let $\varphi$ be a sub-sequence of $\mathbb N$ and let $g_n := f_{\varphi (n)}$. It suffices to prove that $(g_n)$ has a sub-sequence that converges to $f$ in $\sigma(L^p, L^{p'})$. We have $\|g_n-f\|_1 \to 0$, so there is sub-sequence $\psi$ of $\mathbb N$ such that $g_{\psi (n)} -f \to 0$ $\mu$-a.e. or equivalently $g_{\psi (n)} \to f$ $\mu$-a.e.
Brezis' exercise 4.16.1 Let $p \in (1, \infty)$ and $p'$ its Hölder conjugate. Let $f_n,f,g:\Omega \to \mathbb R$ be measurable functions such that $\sup_n \|f_n\|_p < \infty$ and that $f_n \to f$ $\mu$-a.e. Then $f_n \to f$ in the weak topology $\sigma(L^p, L^{p'})$.
Be above part (1.) of the exercise, we get $g_{\psi (n)} \to f$ in $\sigma(L^p, L^{p'})$. This completes the proof.
