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$\lim_{n\rightarrow \infty}\int_0^1f_nhdm=\int_0^1fhdm$, prove $f\in L^p(m)$ , where $1\le p<\infty$.
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When ${f_n}$ is defined on [0,1], $ ||f_n||_p\le1$, $1 , $f$ is integrable on $[0,1]$, $m$ is Lebesgue measure, and $\lim_{n\rightarrow \infty}\int_0^1f_ng \, dm=\int_0^1fg \, dm,$ for any $g\in L^{\infty}(m)$. Prove $f\in L_p$