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- Banach_measure abstract "In mathematics, Banach measure in measure theory may mean a real-valued function on an algebra of all subsets of a set (for example, all subsets of 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 type of generalized measure getting around the phenomenon of non-measurable sets. However, as the Vitali set shows, it cannot be countably additive.A Banach measure on a set Ω is a finite measure μ ≠ 0 on P(Ω), the power set of Ω, such that μ(ω) = 0 for every ω ∈ Ω.A Banach measure on Ω which takes values in the set {0, 1} is called an Ulam measure on Ω. 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.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.".
- Banach_measure wikiPageExternalLink Banach.html.
- Banach_measure wikiPageID "1339598".
- Banach_measure wikiPageLength "1835".
- Banach_measure wikiPageOutDegree "14".
- Banach_measure wikiPageRevisionID "635049499".
- Banach_measure wikiPageWikiLink Algebra_of_sets.
- Banach_measure wikiPageWikiLink Area.
- Banach_measure wikiPageWikiLink Banach_space.
- Banach_measure wikiPageWikiLink Banach–Tarski_paradox.
- Banach_measure wikiPageWikiLink Category:Measures_(measure_theory).
- Banach_measure wikiPageWikiLink Euclidean_plane.
- Banach_measure wikiPageWikiLink Finite_measure.
- Banach_measure wikiPageWikiLink Finitely_additive.
- Banach_measure wikiPageWikiLink Lebesgue_measure.
- Banach_measure wikiPageWikiLink Mathematics.
- Banach_measure wikiPageWikiLink Measure_(mathematics).
- Banach_measure wikiPageWikiLink Measure_theory.
- Banach_measure wikiPageWikiLink Non-measurable_set.
- Banach_measure wikiPageWikiLink Sigma_additivity.
- Banach_measure wikiPageWikiLink Spectral_measure.
- Banach_measure wikiPageWikiLink Spectral_theory_of_ordinary_differential_equations.
- Banach_measure wikiPageWikiLink Two-dimensional_space.
- Banach_measure wikiPageWikiLink Vitali_set.
- Banach_measure wikiPageWikiLink Σ-finite_measure.
- Banach_measure wikiPageWikiLinkText "Banach measure".
- Banach_measure wikiPageWikiLinkText "finitely additive measure".
- Banach_measure hasPhotoCollection Banach_measure.
- Banach_measure wikiPageUsesTemplate Template:Mathanalysis-stub.
- Banach_measure wikiPageUsesTemplate Template:Reflist.
- Banach_measure subject Category:Measures_(measure_theory).
- Banach_measure type Article.
- Banach_measure type Article.
- Banach_measure comment "In mathematics, Banach measure in measure theory may mean a real-valued function on an algebra of all subsets of a set (for example, all subsets of 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 type of generalized measure getting around the phenomenon of non-measurable sets.".
- Banach_measure label "Banach measure".
- Banach_measure sameAs m.04t_1v.
- Banach_measure sameAs Q4853764.
- Banach_measure sameAs Q4853764.
- Banach_measure wasDerivedFrom Banach_measure?oldid=635049499.
- Banach_measure isPrimaryTopicOf Banach_measure.