Topology and Borel Sets

Borel Sets is the smallest cosi-algebra of a topology, so it contains more than the 'open sets' of topology.
In Rudin, Real and Complex Analysis, proof of a lot of theorems start with the 'open sets' and then use the bounding open sets and compact sets to prove for the other sets.
Say, for arbitrary Borel set E, find a compact set KE, s.t. u(V-K)=0. So K and V are actually very tight bondings to the measure of E.
Then this bounding can be used in inequalities. See Riesz Representation Theorem and theorems 2.17, 2.18.