In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

discipline of

measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...

, a Banach measure is a certain type of

content Content or contents may refer to: Media * Content (media), information or experience provided to audience or end-users by publishers or media producers ** Content industry, an umbrella term that encompasses companies owning and providing mas ...

used to formalize geometric area in problems vulnerable to the

axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...

. Traditionally, intuitive notions of area are formalized as a classical,

countably additive In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivity ...

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 Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

finitely additive In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivity ...

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 Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original ...

showed that it is possible to define a Banach measure for the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

, consistent with the usual

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...

. This means that every Lebesgue-measurable subset of

\mathbb^2

is also Banach-measurable, implying that both measures are equal. The existence of this measure proves the impossibility of a

Banach–Tarski paradox The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be p ...

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
{{mathanalysis-stub Measures (measure theory)