How can you prove the following:
Let $(\mu_n)_{n\in \mathbb N}$ a sequence of measures on a measure space $(\Omega, \mathcal F)$, such that $\mu_n \uparrow \mu$ for a measure $\mu$, in the sense that $(\mu_n)$ is monotone ($=$non-decreasing) and $\mu_n \to \mu$ pointwise. Then $\int\limits f\,d\mu_n \uparrow \int\limits f\,d\mu$ for all non-negative and measurable functions $f$.
I seem to be able to verify it for simple functions. But the general case (using the plain monotone convergence theorem) involves switching limits of sequences and I'm not familiar with any conditions that allow this (specifically none that obviously apply here).
Motivation: If $\mu$ is $\sigma$-finite we can define $\mu_n(A):=\mu(A\cap F_n)$, where $F_n \uparrow \Omega$ with $\mu(F_n)<\infty$. Then $(\mu_n)$ is a sequence of finite measures with $\mu_n \uparrow \mu$. So it seems in some cases it may be easier to first prove a statement about finite measures and then extend it through the above theorem.