On $[0,1]$, suppose $\|f_n\|_p\le 1$,
$\lim_{n\rightarrow \infty}\int_0^1f_nh\, dm=\int_0^1fh\, dm$, for any $h\in L^\infty(\mu)$,
I need to prove $f\in L^p(\mu)$ , where $1\le p<\infty$.
I only find out when setting $h(x)=\dfrac{\overline{f(x)}}{|f(x)|}$, it can get $f\in L^1(\mu)$, But I have no idea about other situation.