Outer measure

Remarks

Example: Lebesgue outer measure on $\R$ is $$\lambda^*(A) = \inf \Big\{ \sum_{n} (b_n - a_n) : A \subseteq \bigcup_{n} (a_n, b_n) \Big\},$$ the infimum over all countable covers of $A$ by open intervals. Clause 1 holds since $\varnothing$ is covered by intervals of arbitrarily small total length. Clause 2 holds since a cover of $B$ also covers $A$. Clause 3 follows by covering each $A_n$ within $\varepsilon / 2^n$ of $\lambda^*(A_n)$. Non-example: on $X = \set{1, 2}$, the map with $\mu(\varnothing) = 0$, $\mu(\set{1}) = \mu(\set{2}) = 1$ and $\mu(X) = 3$ is monotone, but it fails clause 3 because $3 > 1 + 1$. Some authors, such as Federer, call outer measures simply measures. The outer measures defined here are the ones to which Carathéodory's criterion applies.