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 is a finite measure on, the power set of, such that for every.
A Banach measure on which takes values in is called an Ulam measure on.
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.
