The problem I'm working on is: Prove that for a family of measurable sets $A_k$ in $[a,b]$ the following is true $$\lim_{k \rightarrow \infty} \mu \left(\bigcup_{n=1}^k A_n \right) = \mu \left(\bigcup_{n=1}^\infty A_n \right)$$
This post was the most similar to my problem, however, the solution doesn't include the union(since it's an increasing sequence?)
Also, the answer to this question includes exactly what I am looking for , but says "there is no reason to think that" the equality always holds.
This got me confused. Please help!