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

Let $$1 and $$\{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 $$g_0 \in L^q(\mathbb{R})$$, $$1 be any function and denote $$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)

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.