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Lebesgue measure

In mathematics, Lebesgue measure is the standard way of assigning a notion of length to subsets of the real line, area to regions of the Euclidean plane, and volume to subsets of Euclidean space in dimensions three and higher. Often denoted λ ( ⋅ ) {\displaystyle \lambda (\cdot )} , it is used throughout mathematical analysis, especially in the…

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Overview

Elementary construction

Carathéodory characterization

Examples

Properties

Null sets

Construction of the Lebesgue measure

Relation to other measures

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Map overview Semantic statistics

Lebesgue measure

Nodes99
Edges98
Triples83
Avg. degree1.98
Density0.020202
Components1

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Lebesgue measure

Top relations

related to Examples · 18
Lebesgue measure → Any, Any Cartesian, Borel, Determinacy, Euler's, However, If, In, Lebesgue, Lebesgue-measurable, Liouville, Moreover, Osgood, Peano, The, The Cantor, Their, Vitali
related to Intuition · 12
Lebesgue measure → Each, If, Intuitively, Lebesgue, Lebesgue-measurable, Such, That, The, The Lebesgue, These, This, Whether
related to Null sets · 12
Lebesgue measure → All, An, Cantor, Euclidean, Hausdorff, Here Hausdorff, If, Lebesgue, Lipschitz, On, Smith, Volterra
related to Construction of the Lebesgue measure · 10
Lebesgue measure → Carathéodory's, Cartesian, Fix, For, It, Lebesgue, Lebesgue-measurable, The, These Lebesgue-measurable, We
related to Properties · 10
Lebesgue measure → Borel, Countable, If, Lebesgue, Lebesgue-measurable, More, Not, Radon, The Lebesgue, This
related to Elementary construction · 8
Lebesgue measure → In, Jordan, Lebesgue, On, Riemann, Similar, The, This
related to Relation to other measures · 6
Lebesgue measure → Borel, Lebesgue, Lebesgue-measurable, The Borel, The Haar, While
is a · 4
Lebesgue measure → application of Carathéodory's extension theorem, Jordan content, length b, standard way of assigning a notion of length to subsets of the real line
see also · 3
Lebesgue measure → FarahLebesgue's, Jordan, Liouville

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Important terminology

measure lebesgue displaystyle sets set lebesgue-measurable mathbb open subset borel intervals lambda countable every textstyle real length zero volume defined

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Lebesgue measureis astandard way of assigning a notion of length to subsets of the real line0.90text
Lebesgue measureis aJordan content0.90text
Lebesgue measureis alength b0.90text
Lebesgue measureis aapplication of Carathéodory's extension theorem0.90text
Lebesgue measurerelated to Construction of the Lebesgue measureThe0.60section
Lebesgue measurerelated to Construction of the Lebesgue measureLebesgue0.60section
Lebesgue measurerelated to Construction of the Lebesgue measureCarathéodory's0.60section
Lebesgue measurerelated to Construction of the Lebesgue measureIt0.60section
Lebesgue measurerelated to Construction of the Lebesgue measureFix0.60section
Lebesgue measurerelated to Construction of the Lebesgue measureCartesian0.60section
Lebesgue measurerelated to Construction of the Lebesgue measureFor0.60section
Lebesgue measurerelated to Construction of the Lebesgue measureWe0.60section

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