Banach measure
Encyclopedia
In mathematics
, Banach measure in measure theory may mean a real-valued function on the algebra of all sets (for example, in the plane), by means of which a rigid, finitely additive area
can be defined for every set, even when a set does not have a true geometric area. That is, this is a theoretical definition getting round the phenomenon of non-measurable set
s. However, as the Vitali set
shows, it cannot be countably additive.
The existence of Banach measures proves the impossibility of a Banach–Tarski paradox
in two dimensions.
The concept of Banach measure is to be distinguished from the idea of a measure taking values in a Banach space
, for example in the theory of spectral measures.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Banach measure in measure theory may mean a real-valued function on the algebra of all sets (for example, in the plane), by means of which a rigid, finitely additive area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
can be defined for every set, even when a set does not have a true geometric area. That is, this is a theoretical definition getting round the phenomenon of non-measurable set
Non-measurable set
In mathematics, a non-measurable set is a set whose structure is so complicated that it cannot be assigned any meaningful measure. Such sets are constructed to shed light on the notions of length, area and volume in formal set theory....
s. However, as the Vitali set
Vitali set
In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by . The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence is proven on the assumption of the axiom of...
shows, it cannot be countably additive.
The existence of Banach measures proves the impossibility of a Banach–Tarski paradox
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set theoretic geometry which states the following: Given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces , which can then be put back together in a different way to yield two...
in two dimensions.
The concept of Banach measure is to be distinguished from the idea of a measure taking values in a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
, for example in the theory of spectral measures.