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, finitely additive measure , defined for every subset of , and whose value is 0 on finite subsets.
A Banach measure on which takes values in } is called an 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. This means that every Lebesgue-measurable subset of is also Banach-measurable, implying that both measures are equal.
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.
References
External links
Stefan Banach bio
Measures (measure theory)