Lebesgue integral over Infinite measure sets

Question

I would apreciate if someone could tell me wether this is true or false, or any advice on how to prove it or disprove it:

Let $f$ be a positive measurable function over $(X,S)$ where S is a $\sigma$-field.

If $\int fd\mu< +\infty $ where $\mu$ is a measure then :

$$\forall E\in S:\mu(E)=+\infty \Rightarrow \int_{E}fd\mu=0$$

Answer 1

Let $X= \mathbb{R}$ with the usual $\sigma$-algebra, then $e^{-|x|}$ is a counterexample.

Answer 2

Assuming that $\mu(X)=\infty$, it is trivially false: just take $E=X$.