Suppose $\mu_1$ and $\mu_2$ are finite and non-negative measures. I am curious if, given those two assumptions, is it always the case that
$\int_{x \in A} f(x) d\mu_1 (x) + \int_{x \in A} f(x) d\mu_2 (x) = \int_{x \in A} f(x) d(\mu_1 (x) + \mu_2 (x))$
ie. Provided the integral bound and integrated function are identical, can we combine measures like this into sums?
Yes.
If $\mu$ is prescribed on measurable sets by $$B\mapsto\mu_1(B)+\mu_2(B)\tag1$$ then $\mu$ is a measure and equality $(1)$ concerning measurable sets can easily be expanded to integrable functions $f$ resulting in:$$f\mapsto\int fd\mu_1+\int fd\mu_2$$
This by going the usual way: characteristic function $\to$ simple function $\to$ nonnegative measurable function $\to$ integrable function.