In the mathematical discipline of measure theory, a Banach measure is a certain way to assign a size (or area) to all subsets of the Euclidean plane, consistent with but extending the commonly used Lebesgue measure. While there are certain subsets of the plane which are not Lebesgue measurable, all subsets of the plane have a Banach measure. On the other hand, the Lebesgue measure is countably additive while a Banach measure is only finitely additive (and is therefore known as a "content").
Stefan Banach proved the existence of Banach measures in 1923.[1] This established in particular that paradoxical decompositions as provided by the Banach-Tarski paradox in Euclidean space R3 cannot exist in the Euclidean plane R2.