TITLE: Convergence in metric and a.e
QUESTION [6 upvotes]: 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?

REPLY [4 votes]: Just to add to the other answers (since it was not explicitly stated): there is a sequence in $L_1[0,1]$ that converges in measure but not pointwise a.e. 
For example: 
$f_1(x)=1$, 
$f_2(x)= \chi_{[0,1/2]}$
$f_3(x)=\chi_{[1/2,1]}$
$f_4(x)=\chi_{[0,1/4]}  $
$f_5(x)=\chi_{[1/4,1/2]}  $
$f_6(x)=\chi_{[1/2,3/4]}  $
$f_7(x)=\chi_{[3/4,1]}  $
$f_8(x)=\chi_{[0,1/8]}  $
$\phantom{f_8(x)}\ \ \vdots$
Where $\chi_A$ is the indicator function on $A$.
$\{f_n\}$ converges to measure to 0 but does not converge pointwise a.e.