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

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.
