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?
Convergence in metric and a.e
6
$\begingroup$
measure-theory
convergence
metric-spaces
-
0But there is always a metric such that convergence in measure iff convergence with respect to the metric. – 2011-12-01
-
3See also [this MO thread](http://mathoverflow.net/questions/5537/notions-of-convergence-not-corresponding-to-topologies) and [this blog post](http://chromotopy.org/?p=354). – 2011-12-01