Let \(X\) be a set and \(S\subset \mathcal {P}(X)\) a collection of subsets of \(X\) containing the empty set and the whole space. Let \(\psi :S\to [0,\infty ]\) be a function such that \(\psi (\emptyset )=0\). Define \(\psi ^*:\mathcal {P}(X)\to [0,\infty ]\) as \[ \psi ^*(A)=\inf \left \{\sum _{i=1}^{\infty }\psi (S_i): A\subseteq \bigcup _{i=1}^{\infty }S_i\right \}. \]
Define what it means to be an outer measure and show that \(\psi ^*\) is an outer measure.