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# Hahn-Kolmogorov theorem

Let $\mathscr{A}_{0}$ be an algebra of subsets of a set $X$. If a finitely additive measure $\mu_{0}\colon\mathscr{A}\to\mathbb{R}\cup\{\infty\}$ satisfies

$\mu_{0}(\bigcup_{{n=1}}^{\infty}A_{n})=\sum_{{n=1}}^{\infty}\mu_{0}(A_{n})$ |

for any disjoint family $\{A_{n}:n\in\mathbb{N}\}$ of elements of $\mathscr{A}_{0}$ such that $\cup_{{n=0}}^{\infty}A_{n}\in\mathscr{A}_{0}$, then $\mu_{0}$ extends to a measure defined on the $\sigma$-algebra $\mathscr{A}$ generated by $\mathscr{A}_{0}$; i.e. there exists a measure $\mu\colon\mathscr{A}\to\mathbb{R}\cup\{\infty\}$ such that its restriction to $\mathscr{A}_{0}$ coincides with $\mu_{0}$.

If $\mu_{0}$ is $\sigma$-finite, then the extension is unique.

Synonym:

Hahn-Kolmogorov extension theorem, Kolmogorov extension theorem

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

28A10*no label found*

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