simultanious convergence of integral norms

Question

Suppose I have a sequence of measurable functions $f_j\in L^p\cap L^q$. I am wondering what can I deduce about the behavior of the sequence in one space based on its behavior in the other.

Specifically, I am interested in the following two questions:

1. If $f_j$ is cauchy with respect to the norm on $L^p$ is it cauchy with respect to the $L^q$ norm as well?
2. If $f_j$ converges in $L^p$ to $g$ and in $L^q$ to $h$, is it true that $g=h$ almost everywhere?

Similar to this question but I am not necessarily interested in the case of $\mu(X)<\infty$

Answer 1

Whether convergence in one $L^p$ implies convergence in another one depends on the ground space on which the functions are defined. The case of finite measure is explained by Hölder's inequality and the case of a purely atomic space is the other way round.

For the other question: convergence in any $L^p$ implies convergence in measure which implies that the limits have to be the same.