Let \(\mu \) be the outer measure on \(\mathbb {R}\) defined by \[ \mu (A) = \inf \left \{ \sum _{i=1}^{\infty }(b_{i} - a_{i}) \mid A \subseteq \bigcup _{i=1}^{\infty }(a_{i}, b_{i}],\ a_{i} \le b_{i} \right \} \]
for every \(A \subset \mathbb {R}\). Prove that \(\mu ((a, b]) = b - a\). You may use that the length of a segment covered by finitely many segments is less than or equal to the sum of the lengths of segments in the covering.