Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 26K / b / Solution

The Caratheodory extension theorem states that a premeasure ${\mu }_0$ on an algebra of subsets of $X$ extends to a measure on the generated sigma-algebra. If ${\mu }_0$ is sigma-finite, this extension is unique. In particular, a finite premeasure has a unique extension.