Banach measure

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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 {0, 1} 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.[1]

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[edit]

  1. ^ Stewart, Ian (1996), From Here to Infinity, Oxford University Press, p. 177, ISBN 9780192832023 .

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