nLab outer measure

Idea

An outer measure is a function that assigns a “size” to every subset of a given set $X$. Unlike a measure, it is not required to be countably additive, but only countably subadditive. It serves as the starting point for the Caratheodory construction.

Definition

Definition

An outer measure on a set $X$ is a function $\mu^* \colon \mathcal{P}(X) \to [0, \infty]$ satisfying the following axioms:

1. Null empty set: $\mu^*(\emptyset) = 0$.

2. Monotonicity: If $A \subseteq B \subseteq X$, then $\mu^*(A) \leq \mu^*(B)$.

3. Countable subadditivity: For any sequence of subsets $\{A_n\}_{n=1}^\infty$ of $X$,

   $$\mu^*\left( \bigcup_{n=1}^\infty A_n \right) \leq \sum_{n=1}^\infty \mu^*(A_n)$$

Construction from pre-measures

The most common way to produce an outer measure is from a pre-measure? $\mu_0$ defined on a ring of sets? (or semi-ring) $\mathcal{R}$ covering $X$. For any $A \subseteq X$, we define:

$$\mu^*(A) = \inf \left\{ \sum_{n=1}^\infty \mu_0(E_n) : A \subseteq \bigcup_{n=1}^\infty E_n, E_n \in \mathcal{R} \right\}$$

Related entries

• measure
• Carathéodory construction?
• measurable set