Convergence in metric and a.e How might I show that there's no metric on the space of measurable functions on $([0,1],\mathrm{Lebesgue})$ such that a sequence of functions converges a.e. iff the sequence converges in the metric?

In fact, the almost converge doesn't correspond to any topology. We use the following fact: 

Let $(X,\mathcal T)$ a topological space. A sequence $\{x_n\}$ converges to $x$ on $(X,\mathcal T)$ if and only if for all subsequence $\{x_{n_k}\}$ we can extract a converging subsequence to $x$. 

Consider a sequence $\{X_n\}$ of random variables which converges in probability but not almost surely to $X$. For each subsequence of $\{X_n\}$, we can extract an almost everywhere converging subsequence, which yield a contradiction.
But there is a metric for the convergence in probability, namely 
$$\delta(X,Y):= \int_{ \Omega}\dfrac{|X(\omega)-Y(\omega)|}{1+|X(\omega)-Y(\omega)|}d\,\mathbb P(\omega).$$
