I have bumped into this problem in the introduction of Measure-theory notes.
Let $(X,\mathcal{M},\mu)$ a finite measure space and let $(A_n){n\in\mathbb{N}}$ , $(B_n){n\in\mathbb{N}}$ two sequences of measurable subsets so that $A_n \subset B_n$ for all $n\in\mathbb{N}$. Prove: $$\mu \left(\bigcup_{n\in \mathbb{N}} B_n \right)-\mu \left(\bigcup_{n\in \mathbb{N}} A_n \right) \leq \sum_{n\in \mathbb{N}}(\mu(B_n)-\mu(A_n))$$
It seems to be consequence of some $\sigma$-aditivity or subaditivity properties (or may be solved considering the sequence of $C_n=B_n - A_n$) but i am struggling to get a clear and complete proof. Thank you in advance.