Convergence both in $L^p$ and $L^q$

Question

I am trying to understand convergence in Lp spaces a little bit better.

If we have $p, q \in [1, \infty]$ and a sequence of functions $(f_n)_{n\in\mathbb{N}} \subset L^p(\mathbb{R}^d) \;\cap\; L^q(\mathbb{R}^d)$ with

• $\Vert f_n - f \Vert_p \rightarrow 0$ for some $f \in L^p$ and
• $\Vert f_n - g \Vert_q \rightarrow 0$ for some $g \in L^q$.

Is it then true that $f = g$ almost everywhere?

I can't think of a good counter example or a proof.

Answer 1

Recall that if $f_n \to f$ in $L^p$ then there is a subsequence along which $f_{n_k} \to f$ almost everywhere. But if $f_n \to f$ in $L^q$ then $f_{n_k} \to f$ in $L^q$ too, and by the same reasoning there is a further subsequence converging to $g$ almost everywhere. Thus $f=g$ almost everywhere.

Answer 2

If you don't mind a powerful theorem, here is a proof using Vitali's convergence theorem. This states that the convergence $f_n\to f$ in $L^p$ is equivalent to three conditions. To show that $f_n\to g$ in $L^p$, we only make reference to $g$ in (i) of the theorem (in the link), and then the other two are given already by $f_n\to f$ in $L^p$. Since $g\in L^p$ already, we have that $f_n\to g$ in $L^p$. Then, we can conclude by almost everywhere uniqueness of $L^p$ limits.