Is a countable intersection of open sets in $\mathbb R$ Lebesgue measurable?

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.

• its complement is a countable union of closed sets – Rolf Hoyer Apr 17 '15 at 22:57
• @RolfHoyer: Yes. – Bear and bunny Apr 17 '15 at 22:59