User:Camillo.delellis/sandbox

2020 Mathematics Subject Classification: Primary: 28A [MSN][ZBL]

Definition

An outer measure is a set function $\mu$ such that

• Its domain of definition is an hereditary $\sigma$-ring (also called $\sigma$-ideal) of subsets of a given space $X$, i.e. a $\sigma$-ring $\mathcal{R}\subset \mathcal{P} (X)$ with the property that for every $E\in \mathcal{R}$ all subsets of $E$ belong to $\mathcal{R}$;
• Its range is the extended real half-line $[0, \infty$];
• $\mu (\emptyset) =0$ and $\mu$ is $\sigma$-subadditive (also called countably subadditive), i.e. for every countable family $\{E_i\}\subset \mathcal{R}$ the following inequality holds:

\[ \mu \left(\bigcup_i E_i\right) \leq \sum_i \mu (E_i)\, . \] The most common outer measures are indeed defined on the full space $\mathcal{P} (X)$ of subsets of $X$.