Let $(X,\mathcal{F},\mu)$ be a measure space. Is there a $\mu$-measurable function $f$ which satisfies the following properties?
0) $f \geq 0$
1) $\int_X f d\mu < \infty$
2) The function is not identically zero
3) $\inf \{f(x) | x \in F \} = 0$, $\forall F \in \mathcal{F}_\mu, F \neq \phi$