Measure theory generalises the intuitive notions of length, area and volume into a rigorous framework for assigning a size to a set, providing the foundation on which modern integration and probability are built. Its modern form was laid out by Emile Borel, Henri Lebesgue, Constantin Caratheodory and others in the late nineteenth and early twentieth centuries. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionWhich subsets of a space can be assigned a consistent, countably additive notion of size, and how does that assignment let integration be defined rigorously over them? 1 Key DebateWhether every subset of the real line can be measured at all: Giuseppe Vitali's 1905 construction of a non-measurable set, sharpened two decades later by the Banach-Tarski paradox, showed that no measure extending ordinary length can consistently be defined on every subset without contradiction. 1 Classification
Pure or AppliedBoth / Interdisciplinary 1 Connections
Associated With
Source Wikipedia: Measure (Mathematics)
Probability, Concepts Probability's own entity-description: the subject "waited until 1933 for a fully rigorous axiomatic footing, when Andrey Kolmogorov placed it within measure theory using three axioms". Already In Branch probability-and-statistics; no measure-theory tie before this write.
Includes
Source Wikipedia: Beppo Levi
Source Wikipedia: Constantin Carathéodory
Source Cramer-Wold theorem (Wikipedia)
Source Falconer's Conjecture (Wikipedia)
Source Giuseppe Vitali (Wikipedia)
Source Gustave Choquet, Biography (MacTutor History of Mathematics)
Source Henri Lebesgue (Wikipedia)
Source Herbert Robbins, Biography (MacTutor History of Mathematics)
Source Wikipedia: Kazimierz Kuratowski
Source Lebesgue-Stieltjes integration (Wikipedia)
Source Wolfram MathWorld: Measure (Mathematics)
Source Set Function (Wikipedia)
Source Structure theorem for Gaussian measures - Wikipedia
Source Vitali covering lemma, Wikipedia
Sources
1. Wikipedia: Measure (Mathematics)
Wikimedia FoundationLead section
The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Carathéodory, and Maurice Fréchet, among others.
Non-measurable sets section
If the axiom of choice is assumed to be true, it can be proved that not all subsets of Euclidean space are Lebesgue measurable
Lead section
Far-reaching generalizations (such as spectral measures and positive operator-valued measures) of measure are widely used in quantum physics and physics in general.
- In Category: Branches of Mathematics
- Associated With: Analysis
View the Source Structure theorem for Gaussian measures - Wikipedia
Includes: Structure Theorem for Gaussian Measures, LeadQuote, Includes: Structure Theorem for Gaussian Measures, Lead
the structure theorem for Gaussian measures shows that the abstract Wiener space construction is essentially the only way to obtain a strictly positive Gaussian measure on a separable Banach space
Vitali covering lemma, Wikipedia
Includes: Vitali Covering Theorem, Lead sentenceQuote, Includes: Vitali Covering Theorem, Lead sentence
mbinatorial and geometric result commonly used in measure theory of Euclidean spaces.
Set Function (Wikipedia)
Includes: Set Function, Lead sentenceQuote, Includes: Set Function, Lead sentence
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and
Cramer-Wold theorem (Wikipedia)
Henri Lebesgue (Wikipedia)
Includes: Henri Lebesgue, Lead paragraph [in-branch]Quote, Includes: Henri Lebesgue, Lead paragraph [in-branch]
his theory of integration
Giuseppe Vitali (Wikipedia)
Includes: Giuseppe Vitali, Lead paragraph [in-branch 2]Quote, Includes: Giuseppe Vitali, Lead paragraph [in-branch 2]
non-measurable subset of real numbers
Wikipedia: Kazimierz Kuratowski
Includes: Kazimierz Kuratowski, LeadQuote, Includes: Kazimierz Kuratowski, Lead
He is primarily known for his contributions to set theory, topology, measure theory and graph theory.
Wikipedia: Constantin Carathéodory
Includes: Constantin Carathéodory, LeadQuote, Includes: Constantin Carathéodory, Lead
He made significant contributions to real and complex analysis, the calculus of variations, and measure theory.
Wikipedia: Beppo Levi
Includes: Beppo Levi, LeadQuote, Includes: Beppo Levi, Lead
Later he proved some foundational results concerning Lebesgue integration, including what is commonly known as Beppo Levi's lemma.
Lebesgue-Stieltjes integration (Wikipedia)
Includes: Lebesgue-Stieltjes Integration, Lead paragraphQuote, Includes: Lebesgue-Stieltjes Integration, Lead paragraph
In measure-theoretic analysis and related branches of mathematics, Lebesgue-Stieltjes integration generalizes both Riemann-Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework.
Wolfram MathWorld: Measure (Mathematics)
Wolfram MathWorldIncludes: Measure (Mathematics), Subject classificationsQuote, Includes: Measure (Mathematics), Subject classifications
Calculus and Analysis Measure Theory
Falconer's Conjecture (Wikipedia)
Includes: Falconer's Conjecture, Lead sectionQuote, Includes: Falconer's Conjecture, Lead section
In geometric measure theory, Falconer's conjecture, named after Kenneth Falconer
Gustave Choquet, Biography (MacTutor History of Mathematics)
Includes: Gustave Choquet, SummaryQuote, Includes: Gustave Choquet, Summary
Gustave Choquet was a French mathematician who worked in functional analysis, potential theory, topology and measure theory.
Herbert Robbins, Biography (MacTutor History of Mathematics)
Includes: Herbert Robbins, SummaryQuote, Includes: Herbert Robbins, Summary
Herbert Robbins was an American mathematician and statistician who worked in topology, measure theory, statistics,and other fields.
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