We say sequence $\displaystyle <f_n>$ in $\displaystyle L^p$converges in$\displaystyle L^p$(sometimes we also call itconverge in mean) to a function $\displaystyle f\in L^p$ if $\displaystyle \lim\limits_{n\to\infty}\|f_n-f\|_p=0$. If $\displaystyle g\in L^p$ and $\displaystyle f=g$ a.e., we have $\displaystyle \|g-f_n\|_p=\|f-f_n\|_p$ so $\displaystyle <f_n>$ converges in $\displaystyle L^p$ also to $\displaystyle g$. From this we see that limit function of convergence in $\displaystyle L^p$ is not necessarily unique. My question is: iffandgare both limit function of convergence in $\displaystyle L^p$, what is the relation betweenfandg? Are they necessarily equal a.e.? If not, could you please come up with a counterexample, that is, $\displaystyle <f_n>$ converges in $\displaystyle L^p$ to bothfandgandfis not equal toga.e.? Thanks!