Let $(X, M, \mu\ )$ be a measure space. Let $f$ be a positive measurable function with $\int_X f d \mu\ < \infty\ .$ Then for every $\epsilon\ > 0$, there is a set $E \in M$ such that $\mu(E)<\infty$ and \begin{equation} \int_X f d \mu\ \leq \int_E f d \mu\ + \epsilon \end{equation}
This problem appears on Bartle's Elements of Integration and Lebesgue Measure. I couldn't prove it for the life of me. And obviously, $E$ has to be a proper subset of $X$, otherwise, this is just trivial. Any idea?