Outer measure

An outer measure ${\mu }^{\ast }$ is defined on every subset, vanishes on the empty set, is monotone, and is countably subadditive.

A set $E$ is measurable for an outer measure ${\mu }^{\ast }$ when every set $A$ is split additively by $E$:The measurable sets form a sigma-algebra, and the restriction of ${\mu }^{\ast }$ to it is a complete measure.

Lebesgue outer measure on $\mathbb{R}$ iswhere the infimum is over countable interval covers. The Caratheodory measurability criterion selects the Lebesgue-measurable sets.