|
In mathematics, Banach measure in measure theory may mean a real-valued function on an algebra of all subsets of a set (for example, all subsets of the plane), by means of which a rigid, finitely additive area can be defined for every set, even when a set does not have a true geometric area. That is, this is a type of generalized measure getting around the phenomenon of non-measurable sets. However, as the Vitali set shows, it cannot be countably additive. A Banach measure on a set Ω is a finite measure μ ≠ 0 on P(Ω), the power set of Ω, such that μ(ω) = 0 for every ω ∈ Ω. A Banach measure on Ω which takes values in the set is called an Ulam measure on Ω. Banach showed that it is possible to define a Banach measure for the Euclidean plane, consistent with the usual Lebesgue measure. The existence of this measure proves the impossibility of a Banach–Tarski paradox in two dimensions: it is not possible to decompose a two-dimensional set of finite Lebesgue measure into finitely many sets that can be reassembled into a set with a different measure, because this would violate the properties of the Banach measure that extends the Lebesgue measure.〔.〕 The concept of ''Banach measure'' is to be distinguished from the idea of a measure taking values in a Banach space, for example in the theory of spectral measures. ==References== 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「Banach measure」の詳細全文を読む スポンサード リンク
|