Approximate the integral of an unsigned measurable function

Question

I've been struggling with this problem.

Let $f\colon X \to [0,+\infty]$ be an unsigned measurable function. Suppose $\int f < \infty$. Prove that for every $\epsilon > 0$ there exists a set $E \in \mathcal{M}$ with $\mu(E) < \infty$ such that $\int f < \epsilon + \int_E f$.

My idea was to approximate $f$ through simple functions, and given that each simple function partitions the domain into finitely many intervals, choose an interval $E$ such that $(domain\setminus E)$ is in the domain of exactly one of the simple functions used to approximate $f$. I am having a hard time trying to formalize this idea. Can you at least tell me if I am on the right track?

Thanks.

Answer 1

Take $E=\{x:0\leq x \leq n,f(x)\leq n\}$. Then $\mu (E) \leq n< \infty$. Also $\int_{E^{c}} fd\mu \to 0$ as $n \to \infty$ (by DCT) because $E^{c}$ decreases to $\{x:f(x)=\infty\}$ (which has measure $0$) as $n$ increases to $\infty$. Hence $\int_{E^{c}} f d\mu <\epsilon$ for $n$ sufficiently large.