Banach measure

In the mathematical discipline of measure theory, a Banach measure is a certain type of content used to formalize geometric area in problems vulnerable to the axiom of choice.

Traditionally, intuitive notions of area are formalized as a classical, countably additive measure. This has the unfortunate effect of leaving some sets with no well-defined area; a consequence is that some geometric transformations do not leave area invariant, the substance of the Banach-Tarski paradox. A Banach measure is a type of generalized measure to elide this problem.

A Banach measure on a set $Ω$ is a finite measure $μ \neq 0$ on $℘(Ω)$ , the power set of $Ω$ , such that $μ({ω}) = 0$ for every $ω \in Ω$ .

A Banach measure on $Ω$ which takes values in ${0, 1$ } is called an Ulam measure on $Ω$ .

As Vitali's paradox shows, Banach measures cannot be strengthened to countably additive ones.

Stefan 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]

References

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

External links

Stefan Banach bio

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

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