Let \(X\) be a set and \(\mathcal {A}\) be an algebra on \(X\). Let \(\rho _0:\mathcal {A}\to [0,\infty ]\) be a pre-measure on \(\mathcal {A}\). Recall that \[ \rho ^*(S)=\inf \left \{\sum _{i=1}^{\infty }\rho _0(A_i): A_i\in \mathcal {A}\text { for all }i\text { and }S\subset \bigcup _{i=1}^{\infty }A_i\right \} \]
is an outer measure. Define what it means that \(\rho _0\) is a pre-measure and show that if \(A\in \mathcal {A}\) then \(\rho ^*(A)=\rho _0(A)\).