$L^{2}$ convergent, subsequence, directed family of points I have a question about a convergence.
Let $(E,\mathcal{B},m)$ be a measure space. I think the following assertion is very famous:
Let $f_{n},f \in L^{2}(E;m)\quad(n=1,2,\cdots)$. If $f_{n}\to f $  in $L^{2}(E;m)$ then there exists subsequence $(f_{n_{k}})_{k=1}^{\infty}$ of $(f_{n})_{n=1}^{\infty}$ such that $f_{n_{k}}\to f$  $m$-a.e.
By the way, let $(f_{t})_{t>0} \subset L^{2}(E,m),\,f \in L^{2}(E;m)$. We suppose that  $f_{t} \to f \,(t \to 0) $ in $L^{2}(E;m)$. 
In above situation, can we conclude $f_{t} \to f$ $m$-a.e.?

No. All you can conclude is that for every sequence $\{t_n\}$, $t_n\to0$, there is a subsequence $\{t_{n_k}\}$ such that $f_{t_{n_k}}\to f$ $m$-a.e.
