If the answer is yes, how to prove that? Otherwise how to find a counterexample?
Update:
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.
Sigma algebras are closed under countable intersections. Combine this with the fact that Borel measurable sets are also Lebesgue measurable is sufficient to answer your question.