I been reading This, and I want to find some function on those case they are not inclusion each other. They give an example for $q<p$, $(x+1)^{-1/p}$ is a function in $L^q(0,\infty)$ but not in $L^p(0,\infty)$.

So I think there should be also some function $f\in L^p(0,\infty)$ but $f\not \in L^q(0,\infty)$ given $q<p$. But I failed to construct one.