## Theorem 7.3.8: Borel Sets are Measurable

The collection of Borel sets is the smallest sigma-algebra which contains all of
the open sets. Every Borel set, in particular every open and closed set, is
measurable.

This follows from the fact that open sets and closed sets are measurable,
as we have just proved, and so are countable unions (and intersections) of
those sets. Therefore the collection of all measurable sets is a sigma-algebra.
But then, since by definition the Borel sets are the *smallest* sigma
algebra containing the open sets, it follows that the Borel sets are a subset
of all measurable sets and are therefore measurable.