TITLE: Real Analysis Convergence question
QUESTION [8 upvotes]: My question is: For $f_n, g, h \in L^p(X)$, where $X$ is a finite measure space,  if $f_n$ converges to $g$ a.e and $f_n$ converges to $h$ weakly in $L^p$, can we conclude any relationship between $g$ and $h$? Thanks!

REPLY [6 votes]: I will assume $1<p<\infty$. Since we work on a finite measure space, we can use Egoroff's theorem. Fix $\phi\in L^q$ where $q$ is the conjugate exponent (i.e. $p^{-1}+q^{-1}=1$). Fix $\varepsilon>0$, and $\delta>0$ such that if $\mu(A)\leq \delta$ then $\int_A |\phi|^q\leq \varepsilon^q$. Thanks to Egoroff's theorem, we can find $A_{\delta}$ such that $f_n\to g$ uniformly on $A_{\delta}$ and $\mu(A_{\delta}^c)\leq \delta$. So 
\begin{align*}
\left|\int_X(g-h)\phi ~d\mu\right|&\leq \limsup_n \left(\left|\int_X(g-f_n)\phi ~d\mu \right| +\left|\int_X(f_n-h)\phi ~ d\mu \right| \right)\\\
&\leq \limsup_n\left|\int_X(g-f_n)\phi ~ d\mu \right|\\\
&\leq \limsup_n\left(\left|\int_{A_{\delta}}(g-f_n)\phi ~ d\mu \right|+\left|\int_{A_{\delta}^c}(g-f_n)\phi ~d\mu \right|\right)\\\
&\leq \limsup_n\:\sup_{A_{\delta}}|g-f_n|\left(\int_X|\phi|^qd\mu\right)^{\frac 1q}+\sup_k\lVert g-f_k\rVert_{L^p}\left(\int_{A_{\delta}}|\phi|^q\right)^{\frac 1q}\\\
&\leq \varepsilon\cdot \sup_k\lVert g-f_k\rVert_{L^p}
\end{align*} 
so $\int_X(g-h)\phi d\mu=0$ for all $\phi \in L^q$: we conclude that $g=h$ almost everywhere.