I have some set $A$ of Lebesgue measure $\mu(A)=1$. Does this imply that there is some measurable function $f: \mathbb{R}^n \to \mathbb{R}$ such that $\int_A |f| d\mu< \infty, \int_A |f|^2 d\mu= \infty$
Certainly if $A=(0,1]$ we could take something like $f(x)=\frac{1}{\sqrt{x}}$. Is there a canonical way to solve this for arbitrary $A$?