Measure and topology

Abstract

Given a topological space X, there is a natural $\sigma$-algebra of subsets of X, namely the $\sigma$-algebra B _X of Borel sets. When X is locally compact (that is, every point has a relatively compact neighborhood) another useful $\sigma$-algebra is the $\sigma$-algebra generated by the compact G _δ subsets of X. This collection is called the $\sigma$-algebra of Baire sets and is denoted Ba _X . The $\sigma$-algebra Ba _X is defined the same way for arbitrary topological spaces, but it is not so useful when X is not locally compact.