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, finitely additive measure μ ≠ 0, defined for every subset of ℘(Ω), and whose value is 0 on finite subsets.

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. This means that every Lebesgue-measurable subset of ${\displaystyle \mathbb {R} ^{2}}$ is also Banach-measurable, implying that both measures are equal.[1]

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.[2]