If the answer is yes, how to prove that? Otherwise how to find a counterexample?
I've figured out the tricks inside.
A countable intersection of open sets in $\mathbb R$ is equivalent to a countable union of closed sets. Since a countable union of a sequence of pairwise disjoint measurable sets is again measurable, will have a countable union of a sequence of any measurable sets measurable. Since a closed set is measurable, apparently, a countable intersection of open sets in $\mathbb R$ is measurable.
P.S. I'm not sure why this question is off-topic. However, it will help a real variable novice make clear sense of the Borel σ-algebra and the Borel sets.