We say sequence inconverges in(sometimes we also call itconverge 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: iffandgare both limit function of convergence in , what is the relation betweenfandg? Are they necessarily equal a.e.? If not, could you please come up with a counterexample, that is, converges in to bothfandgandfis not equal toga.e.? Thanks!