Relation between a.s. convergence and weak convergence in in $L^p$ spaces Is there a relation between almost-sure convergence and weak convergence of $f_n\in L^p(\mathbb{R})$? (i.e. convergence if tested with $L^{p'}$)
I know that none implies the other (see masses drifting away, or oscillating functions, respectively), but is there an additional condition to add to one to get the other?
For example, in $C(K)$, weak convergence is pointwise convergence plus bounded norms, and my question is if there is a similar result for $L^p$.

Let $(f_n)_{n\geqslant 1}$ be a sequence with converges almost everywhere to $0$. 
In this thread, it is shown that if $\sup_n\lVert f_n\rVert_p$ is finite, then $f_n\to 0$ weakly in $L^p$. Since boundedness in $L^p$ is a necessary condition for the weak convergence in $L^p$, boundedness in $L^p$ is a necessary and sufficient condition for an almost everywhere converging sequence. 
Even strong convergence in $L^p$ does not imply almost everywhere convergence, but if $\sum_n\lVert f_n\rVert_p$ converges, we have $f_n\to 0$ a.e. I am not sure this is what the OP wanted. 
