This is a theorem I'm trying to prove in Royden(4th ed).
Let $(X,\mathcal{M},\mu)$ be a measure space and $f$ be integrable over $X$.
- If $\{X_n\}_{n=1}^\infty$ is an ascending countable collection of measurable subsets of $X$ whose union is $X$, then $ \int_X f~d\mu = \lim_{n\to \infty} \int_{X_n} f~d\mu.$
- If $\{X_n\}_{n=1}^\infty$ is an descending countable collection of measurable subsets of $X$ then $ \int_{\bigcap_{n=1}^{\infty}X} f~d\mu = \lim_{n\to \infty} \int_{X_n} f~d\mu.$
The book says it follows from this theorem: Suppose $f$ is integrable on $X$, $\{X_n\}_{n=1}^\infty$, a disjoint countable colletction of measurable sets whose union is $X$. Then $\int _X f~d\mu = \sum _{n=1}^\infty \int_{X_n} f~d\mu.$ I however can't see it, and I ask for your help.