Let $E\subset \Bbb R^n$ be measurable, $mE>0$. and $0<p_0\neq q_0<\infty$, $L^{p_0}(E)\subset L^{q_0}(E)$. Show $q_0<p_0$.

Question

Let $E\subset \Bbb R^n$ be measurable, $mE>0$. and $0<p_0\neq q_0<\infty$, $L^{p_0}(E)\subset L^{q_0}(E)$. Show $q_0<p_0$.

If $E=(0,\infty)$, we may argue by contradition and use examples like $x^{-a}$ for some $a$. However, for general $E$ with $mE>0$, how to find an example?