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| In [[mathematics]], the '''Hahn–Kolmogorov theorem''' characterizes when a [[finitely additive]] [[function (mathematics)|function]] with [[negative and positive numbers|non-negative]] (possibly infinite) values can be extended to a ''bona fide'' [[measure (mathematics)|measure]]. It is named after the [[Austria]]n [[mathematician]] [[Hans Hahn (mathematician)|Hans Hahn]] and the [[Russia]]n/[[Soviet Union|Soviet]] mathematician [[Andrey Kolmogorov]].
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| ==Statement of the theorem==
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| Let <math>\Sigma_0</math> be an [[Field of sets|algebra of subset]]s of a [[Set (mathematics)|set]] <math>X.</math> Consider a function
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| :<math>\mu_0\colon \Sigma_0 \to[0,\infty]</math>
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| which is ''finitely additive'', meaning that
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| : <math>\mu_0(\bigcup_{n=1}^N A_n)=\sum_{n=1}^N \mu_0(A_n)</math>
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| for any positive [[integer]] ''N'' and <math>A_1, A_2, \dots, A_N</math> [[disjoint set]]s in <math>\Sigma_0</math>.
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| Assume that this function satisfies the stronger ''sigma additivity'' assumption
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| :<math> \mu_0(\bigcup_{n=1}^\infty A_n) = \sum_{n=1}^\infty \mu_0(A_n)</math>
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| for any disjoint family <math>\{A_n:n\in \mathbb{N}\}</math> of elements of <math>\Sigma_0</math> such that <math>\cup_{n=1}^\infty A_n\in \Sigma_0</math>. (Functions <math>\mu_0</math> obeying these two properties are known as [[pre-measures]].) Then,
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| <math>\mu_0</math> extends to a measure defined on the [[sigma-algebra]] <math>\Sigma</math> generated by <math>\Sigma_0</math>; i.e., there exists a measure
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| :<math>\mu \colon \Sigma \to[0,\infty]</math>
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| such that its [[Restriction (mathematics)|restriction]] to <math>\Sigma_0</math> coincides with <math>\mu_0.</math>
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| If <math>\mu_0</math> is <math>\sigma</math>-finite, then the extension is unique.
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| ==Non-uniqueness of the extension==
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| If <math>\mu_0</math> is not <math>\sigma</math>-finite then the extension need not be unique, even if the extension itself is <math>\sigma</math>-finite.
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| Here is an example:
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| We call ''rational closed-open interval'', any subset of <math>\mathbb{Q}</math> of the form <math>[a,b)</math>, where <math>a, b \in \mathbb{Q}</math>.
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| Let <math>X</math> be <math>\mathbb{Q}\cap[0,1)</math> and let <math>\Sigma_0</math> be the algebra of all finite union of rational closed-open intervals contained in <math>\mathbb{Q}\cap[0,1)</math>. It is easy to prove that <math>\Sigma_0</math> is, in fact, an algebra. It is also easy to see that every non-empty set in <math>\Sigma_0</math> is infinite.
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| Let <math>\mu_0</math> be the counting set function (<math>\#</math>) defined in <math>\Sigma_0</math>.
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| It is clear that <math>\mu_0</math> is finitely additive and <math>\sigma</math>-additive in <math>\Sigma_0</math>. Since every non-empty set in <math>\Sigma_0</math> is infinite, we have, for every non-empty set <math>A\in\Sigma_0</math>, <math>\mu_0(A)=+\infty</math>
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| Now, let <math>\Sigma</math> be the <math>\sigma</math>-algebra generated by <math>\Sigma_0</math>. It is easy to see that <math>\Sigma</math> is the Borel <math>\sigma</math>-algebra of subsets of <math>X</math>, and both <math>\#</math> and <math>2\#</math> are measures defined on <math>\Sigma</math> and both are extensions of <math>\mu_0</math>.
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| ==Comments==
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| This theorem is remarkable for it allows one to construct a measure by first defining it on a small algebra of sets, where its sigma additivity could be easy to verify, and then this theorem guarantees its extension to a sigma-algebra. The proof of this theorem is not trivial, since it requires extending <math>\mu_0</math> from an algebra of sets to a potentially much bigger sigma-algebra, guaranteeing that the extension is unique (if <math>\mu_0</math> is <math>\sigma</math>-finite), and moreover that it does not fail to satisfy the sigma-additivity of the original function.
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| == See also ==
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| * [[Carathéodory's extension theorem]]
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| * [[pre-measure]]
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| {{PlanetMath attribution|id=5409|title=Hahn–Kolmogorov theorem}}
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| {{DEFAULTSORT:Hahn-Kolmogorov theorem}}
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| [[Category:Theorems in measure theory]]
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