Counterexample of $L^p(E)$ space when $m(E) = \infty$

Question

Here is the proposition

When $m(E)<\infty$, if $1 \leq p_2 < p_1 < \infty$, then $$ L^{\infty}(E) \subset L^{p_1}(E) \subset L^{p_2}(E) $$

Is the condition $m(E) < \infty$ necessary? If yes, are there any counterexamples when $m(E) = \infty$?