I want to gain a better understanding of the convergence of a sequence of functions $(u_n:I\to \mathbb R)_{n\in \mathbb N}$ in different norms, I know some results, for example that if $u_n(x)\to u(x)$ pointwise a.e. and $u_n\to v$ in $L^1(I)$, then $v(x)=u(x)$ a.e.
However, I don't know if there is a rule of thumb to compare the limits in different conditions, specially when dealing with weak topologies. In particular, I am interest the most in these two scenarios:
Suppose $(u_n)_{n\in \mathbb N}$ is a sequence of continuous functions in $L^\infty(I)$ such that $u_n(x)\to u(x)$ a.e. and $u_n\rightharpoonup v$ weakly in $L^1(I)$, is it true that $v(x)=u(x)$ a.e. ?
If $(u_n)_{n\in \mathbb N}\subset L^p(I)\cap L^q(I)$, where $\frac 1 p+\frac 1 q = 1$ and $(u_n)$ converges strongly in $L^p(I)$ and weakly in $L^q(I)$, does the limits coincide?
In general, if some of these kind of results are false, what kind of 'illness' need the sequence have to have different limits in different norms.
It suffices to treat the case where $I$ is a finite interval; if we treat the case of $I\cap [-N,N]$ for any $N$ we will get $u=v$ almost everywhere on this interval. We can also assume that $u=0$ since the assumption of weak convergence implies that $\lVert u_n\rVert_1$ is bounded which implies, by Fatou's lemma, that $u$ belongs to $L^1$. using Egoroff's theorem, we split for each integer $k$ the set $I$ into $U_k$ and $R_k$, where $U_k$ is such that $\sup_{x\in U_k}\lvert u_n(x)\rvert\to 0$ as $n$ goes to infinity and $\lambda(R_k)<1/k$. Then for each measurable subset of $I$, $$\int_{A}u_n=\int_{U_k\cap A}u_n+\int_{R_k\cap A}u_n.$$ Taking the limit in $n$, we get $\int_A v=\int_{R_k\cap A}v$ and since $v$ is integrable, we get $\int_A v=0$ hence $v=0$ almost everywhere.
The limits coincide, as you can test weak convergence on indicator functions of sets of finite measure.