词条 | Hahn–Kolmogorov theorem |
释义 |
In mathematics, the Hahn–Kolmogorov theorem characterizes when a finitely additive function with non-negative (possibly infinite) values can be extended to a bona fide measure. It is named after the Austrian mathematician Hans Hahn and the Russian/Soviet mathematician Andrey Kolmogorov. Statement of the theoremLet be an algebra of subsets of a set Consider a function which is finitely additive, meaning that for any positive integer N and disjoint sets in . Assume that this function satisfies the stronger sigma additivity assumption for any disjoint family of elements of such that . (Functions obeying these two properties are known as pre-measures.) Then, extends to a measure defined on the sigma-algebra generated by ; i.e., there exists a measure such that its restriction to coincides with If is -finite, then the extension is unique. Non-uniqueness of the extensionIf is not -finite then the extension need not be unique, even if the extension itself is -finite. Here is an example: We call rational closed-open interval, any subset of of the form , where . Let be and let be the algebra of all finite union of rational closed-open intervals contained in . It is easy to prove that is, in fact, an algebra. It is also easy to see that the cardinal of every non-empty set in is . Let be the counting set function () defined in . It is clear that is finitely additive and -additive in . Since every non-empty set in is infinite, we have, for every non-empty set , Now, let be the -algebra generated by . It is easy to see that is the Borel -algebra of subsets of , and both and are measures defined on and both are extensions of . CommentsThis 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 from an algebra of sets to a potentially much bigger sigma-algebra, guaranteeing that the extension is unique (if is -finite), and moreover that it does not fail to satisfy the sigma-additivity of the original function. See also
1 : Theorems in measure theory |
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