If $f_n \to f$ a.e. , and $f_n$ bounded sequence ,then $f_n \to f$ weakly in $L^p(\mathbb{R})$ .

Question

Let $1<p<\infty$ and $\{f_n\}\in L^p(\mathbb{R})$ be a uniformly bounded sequence, i.e. $\|f_n\|_p\le M, \forall n$, for some $M>0$. If $f_n \to f$ a.e. , prove that $f_n \to f$ weakly in $L^p(\mathbb{R})$ .

My attempt :

Let $A$ be any measurable subset $A \subset\mathbb{R}$, by lemma 2 we have $\lim \int_A f_n = \int_A f$ and using lemma 1 we have $f_n \to f$ weakly in $L^p(\mathbb{R})$.

lemma 1: if $1\le p<\infty$ and $\{f_n\}\in L^p(\mathbb{R})$ be a bounded sequence, then $f_n \to f$ weakly in $L^p(\mathbb{R})$ iff for every measurable subset $A \subset\mathbb{R}$, $$\lim \int_A f_n = \int_A f$$ proof:

WLOG that $m(A)<\infty$ , let $g_0 \in L^q(\mathbb{R})$, $1<q\le\infty$ be any function and denote $g= \chi_{A}$;

\begin{align} \int_{\mathbb{R}} g_0f_n - \int_{\mathbb{R}} g_0f & = \int_{\mathbb{R}} (g_0 -g)(f_n-f) + \int_{\mathbb{R}} g(f_n-f) \\ & \le \|g_0-g\|_q\|f_n-f\|_p + \frac{\epsilon}{2} \\ & \le \frac{\epsilon}{2}+\frac{\epsilon}{2} =\epsilon \implies f_n \rightharpoonup f \end{align}

where I used boundedness in the last inequality above.

lemma 2: if $A$ is measurable and $\{f_n\}\in L^p(\mathbb{R})$ be a bounded sequence, then $\{f_n\}$ is uniformly integrable over $A$. (proof in royden ch.7)

Answer 1

Your attempt is almost good. The only gap is that in general, you cannot make for any function $g_0$ the norm $\lVert g_0-\mathbf 1_A\rVert_q$ as small as you wish. However, we can approximate $g_0$ in $L^q$ by a finite linear combination of indicator functions.