Set $(\Omega, \Sigma ,\mu)$ a finite measure space and let $1\leq p <+\infty$. Let $\psi$ be continuous on $\mathbb R$ to $\mathbb R$, and $f \in L_p(\Omega,\mu)$. Show that there is $\psi$ and $f$ such that $\psi \circ f \notin L_p (\Omega,\mu)$.
This question is in an exercise list in my Measure Theory course. I have spent quite sometime trying to come up with the example, but was not able to.
Take $\Omega = [0,1]$ with the usual Lebesgue measure. The function $g(x)=x^r$ is in $L^1$ if and only if $r>-1$.
From the above, we can see that $f(x) = x^{-\frac 1{2p}}$ is in $L^p$ since $|f|^p = x^{-1/2}$ but if we take $\psi(t)=t^3$, then $$ |\psi(f(x))|^p = x^{-3/2} $$ so $\psi\circ f \notin L^p(\Omega)$.