Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 16 / 5 / ii

For a Borel set $B$ and $ϵ>0$, inner regularity supplies a compact set $K\subseteq B$ with $\mu (B)-\mu (K)<ϵ/2$. Independently, outer regularity supplies an open set $U\supseteq B$ with $\mu (U)-\mu (B)<ϵ/2$. Because $K\subseteq U$ and the measure is finite,This creates an arbitrarily small measurable transition region between a compact inner approximation and an open outer approximation. The cutoff in the next step then approximates $1_B$, rather than merely approximating the number $\mu (B)$.