Let $E$ be a measurable set and $1\le p <\infty$.Suppose $\{f_n\}$ is a bounded sequence in $L^P(E)$ and $f \in L^P(E)$ . Then $f_n\rightharpoonup f$ in $L^P(E)$ iff for every measurable subset $A$ of $E$ $$\lim_{n\to \infty}\int_A f_n d\mu= \int_A f d\mu$$
This is Thm 10 ch.8 in Royden and Fitzpatrick, but has no proof. is the following correct?
I guess This $\Rightarrow $ is obvious by letting $g\in L^q$ to be a characteristic function. then $\lim_{n\to \infty}\int_E f_n \chi_A d\mu= \int_E f\chi_A d\mu$.
but about $\Leftarrow $ direction; Suppose $E$ has finite measure and $p>1$.
I tried let $g_0\in L^q$ be any function. Then; \begin{align} \int_E g_0f_n d\mu- \int_E g_0f d\mu & \le \int_E (g_0-\chi_A)(f_n-f) d\mu+ \int_E (f_n-f)\chi_A d\mu \\ & \le \|g_0-\chi_A\|_q\|f_n-f\|_p+ \frac{\epsilon}{2} ~~~~ \small{\text{(1-Holder 2-by the above assumption)}}\\ &\le \frac{\epsilon}{2} + \frac{\epsilon}{2} =\epsilon \\ \end{align} where the last inequality follows from the since $g_0\in L^q$, and simple functions are dense in $L^q$, there exists $\chi_A$ such that $\|g_0-\chi_A\|_q\|f_n-f\|_p \le \frac{\epsilon}{2}$. $\square$
for $p=1$, and if $A$ is not finite measure I don't know how to prove this.