Suppose that $\left(X, \mathcal E, \mu\right)$ is $\sigma$-finite. So we have that $X=\bigcup_{n}A_n$ where $\left(A_n\right)_n$ is an increasing sequence of elements $A_n\in \mathcal E$ where $\mu\left(A_n\right)<\infty$.
My question is then, what is the reason for that you can't switch the limit of the following: $$\lim_{n\rightarrow\infty}\lim_{p\rightarrow\infty}||\textbf{1}_{A_n}f||_p=||f||_\infty$$
Thanks in advance.