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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!

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This may be a sledgehammer and is only a partial answer for the case when $1<p<\infty$.

Here are three facts:

1) Weakly convergent sequences are norm bounded (see for instance page 255 of this ).

2) For $1<p<\infty$, norm bounded sequences that converge pointwise a.e. are weakly convergent to their pointwise limit (see, e.g, Theorem 13.44 in Hewitt and Stromberg’s, *Real and Abstract Analysis*).

3)Weak limits are unique.

Thus, if $f_n$ converges weakly to $h$ in $L_p$ for $1<p<\infty$ and converges pointwise to $g$ a.e, it follows that $f=g$ almost everywhere.

More sledgehammers:

For $p=1$ with a finite measure space:

Weak convergence of $\{f_n\}$ insures that $\{f_n\}$ is uniformly integrable. This is known as the

Dunford-Pettis theorem

(see, e.g., pg. 59 here or IV.8.11 in Dunford and Schwartz’ *Linear Operators, Part 1, General Theory*). By the Vitali Convergence Theorem (see also page 163

here), then, $\{f_n\}$ converges to $g$ in $L_1$. Then, since convergence in norm to $g$ implies weak convergence to $g$, we must have $g=h$ a.e. .

I will assume $1<p<\infty$.Since we work on a finite measured space, we can use Egoroff 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\right|d\mu&\leq \limsup_n\left|\int_X(g-f_n)\phi\right|d\mu+\left|\int_X(f_n-h)\phi\right|d\mu\\

&\leq \limsup_n\left|\int_X(g-f_n)\phi\right|d\mu\\

&\leq \limsup_n\left|\int_{A_{\delta}}(g-f_n)\phi\right|d\mu+\left|\int_{A_{\delta}^c}(g-f_n)\phi\right|d\mu\\

&\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.

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