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!