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In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological space ''X'', the collection of all Borel sets on ''X'' forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on ''X'' is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets). Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory. In some contexts, Borel sets are defined to be generated by the compact sets of the topological space, rather than the open sets. The two definitions are equivalent for many well-behaved spaces, including all Hausdorff σ-compact spaces, but can be different in more pathological spaces. == Generating the Borel algebra == In the case ''X'' is a metric space, the Borel algebra in the first sense may be described ''generatively'' as follows. For a collection ''T'' of subsets of ''X'' (that is, for any subset of the power set P(''X'') of ''X''), let * be all countable unions of elements of ''T'' * be all countable intersections of elements of ''T'' * Now define by transfinite induction a sequence ''Gm'', where ''m'' is an ordinal number, in the following manner: * For the base case of the definition, let be the collection of open subsets of ''X''. * If ''i'' is not a limit ordinal, then ''i'' has an immediately preceding ordinal ''i − 1''. Let *: * If ''i'' is a limit ordinal, set *: The claim is that the Borel algebra is ''G''ω1, where ω1 is the first uncountable ordinal number. That is, the Borel algebra can be ''generated'' from the class of open sets by iterating the operation : to the first uncountable ordinal. To prove this claim, note that any open set in a metric space is the union of an increasing sequence of closed sets. In particular, complementation of sets maps ''Gm'' into itself for any limit ordinal ''m''; moreover if ''m'' is an uncountable limit ordinal, ''Gm'' is closed under countable unions. Note that for each Borel set ''B'', there is some countable ordinal α''B'' such that ''B'' can be obtained by iterating the operation over α''B''. However, as ''B'' varies over all Borel sets, α''B'' will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is ω1, the first uncountable ordinal. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Borel set」の詳細全文を読む スポンサード リンク
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