Math 6210 • Real Analysis

Let \((X, \mathcal {T})\) be a topological space and \(\mathcal {B}\) be its Borel \(\sigma \)-algebra. Show that if \((X, \mathcal {B}, \mu )\) is a \(\sigma \)-finite measure space that is outer regular on finite measure, measurable subsets, then it is automatically outer regular on all measurable subsets. Recall that \(\mu \) is called outer regular on a collection of subsets \(\mathcal {S}\) if for every \(A \in \mathcal {S}\) and \(\epsilon > 0\) there is an open set \(U\) so that \(A \subset U\) and \(\mu (U \setminus A) < \epsilon \).