Off the top of my head, they are equal a.e. Incidentally, the limit functions are unique if instead of talking about functions, you talk about equivalence classes of functions which are equal a.e. See Royden 3rd Ed., pages 118, 119.
We say sequence in converges in (sometimes we also call it converge in mean) to a function if . If and a.e., we have so converges in also to . From this we see that limit function of convergence in is not necessarily unique. My question is: if f and g are both limit function of convergence in , what is the relation between f and g? Are they necessarily equal a.e.? If not, could you please come up with a counterexample, that is, converges in to both f and g and f is not equal to g a.e.? Thanks!