I'm trying to understand why defining a set function $\mu$ by setting $\mu(E) = \int_E f d \mu$ defines a measure, where $f$ is some integrable function.
I know that if $E_1, E_2$ are disjoint, then $\int_{E_1 \cup E_2} f d \mu = \int_{E_1} f d \mu + \int_{E_2} f d \mu$, and that we can extend this to a finite union of pairwise disjoint sets. But how do we get $\sigma$-additvity from this? Do we need to use the monotone convergence theorem or something?
Sure, you can use monotone convergence to see this. Just let $f_n = f\sum_{k=1}^n1_{E_k}.$ (Note, you omitted the fact that $f$ must be non-negative.)