# Thread: limit function of convergence in L^p

1. ## limit function of convergence in L^p

We say sequence $\displaystyle <f_n>$ in $\displaystyle L^p$ converges in $\displaystyle L^p$(sometimes we also call it converge 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: if f and g are both limit function of convergence in $\displaystyle L^p$, 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, $\displaystyle <f_n>$ converges in $\displaystyle L^p$ to both f and g and f is not equal to g a.e.? Thanks!

2. 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.

3. Yes, if f and g are both limit functions of convergence in $\displaystyle L^p$, then f must be equal to g a.e.. This is because convergence in $\displaystyle L^p$ implies convergence in measure and all limit functions of convergence in measure are equal a.e.. Thank myself