Assume that \(\mu :\mathcal {B}_{\mathbb {R}^{4}}\to [0,\infty ]\) is a measure which is outer regular and \(\sigma \)-finite. Show that if \(\mu \) is inner regular on open sets then it is inner regular on all Borel sets. Recall that a Borel measure on \(\mathbb {R}^{4}\) is called outer regular if for any \(A\in \mathcal {B}_{\mathbb {R}^{4}}\) we have \[ \mu (A)=\inf \{\mu (U): U\text { is open and }A\subset U\} \]
and it is called inner regular if
\[ \mu (A)=\sup \{\mu (K): K\text { is compact and }K\subset A\}. \]