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Branches of Mathematic

Measure Theory

Also Known As Lebesgue Measure Theory
Analysis

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 Question
Which 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 Debate
Whether 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 Applied
Both / Interdisciplinary 1
Connections

Associated With

Analysis, Branches of Mathematics
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

Cramer-Wold Theorem, Theorems
Source Cramer-Wold theorem (Wikipedia)
Giuseppe Vitali, Mathematicians
Source Giuseppe Vitali (Wikipedia)
Henri Lebesgue, Mathematicians
Source Henri Lebesgue (Wikipedia)
Lusin's Theorem, Theorems
Set Function, Mathematical Objects
Source Set Function (Wikipedia)
Vitali Covering Theorem, Theorems
Source Vitali covering lemma, Wikipedia
Sources
1. Wikipedia: Measure (Mathematics)
Wikimedia Foundation
  • Introduction
    The foundations of modern measure theory were laid in the works of Emile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Caratheodory, and Maurice Frechet
  • 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, pure or applied classification
    Far-reaching generalizations of measure are widely used in quantum physics and physics in general.
  • In Category: Branches of Mathematics
  • Associated With: Analysis
View the Source
Vitali covering lemma, Wikipedia
Includes: Vitali Covering Theorem, Lead sentence
Quote, Includes: Vitali Covering Theorem, Lead sentence
mbinatorial and geometric result commonly used in measure theory of Euclidean spaces.
View the Source
Set Function (Wikipedia)
Includes: Set Function, Lead sentence
Quote, 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
View the Source
Cramer-Wold theorem (Wikipedia)
Includes: Cramer-Wold Theorem, Lead sentenceView the Source
Henri Lebesgue (Wikipedia)
Includes: Henri Lebesgue, Lead paragraph [in-branch]
Quote, Includes: Henri Lebesgue, Lead paragraph [in-branch]
his theory of integration
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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
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