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Measure theory

Table of contents

Definitions of mathematical integration

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Lp spaces

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Measures (measure theory)

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Measures (set theory)

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Probability theory

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Theorems in measure theory

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Absolute continuity

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Abstract Wiener space

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Almost everywhere

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Atom (measure theory)

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Aubin–Lions lemma

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Ba space

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Baire set

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Borel isomorphism

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Brezis–Lieb lemma

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Cantor set

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Carathéodory's criterion

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Cartan–Hadamard conjecture

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Clarkson's inequalities

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Coarea formula

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Concentration of measure

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Continuity set

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Conull set

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Convergence in measure

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Convergence of measures

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Curvature of a measure

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Cylinder set

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Distribution function (measure theory)

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Equivalence (measure theory)

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Essential infimum and essential supremum

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Essential range

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Euler measure

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Finite-dimensional distribution

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Fuzzy measure theory

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Hanner's inequalities

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Hausdorff density

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Hausdorff paradox

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Homological integration

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Indicator function

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Infinite-dimensional Lebesgue measure

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Intensity (measure theory)

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Klee's measure problem

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Laplacian of the indicator

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Lebesgue integration

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Lifting theory

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Littlewood's three principles of real analysis

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Locally integrable function

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Loeb space

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Lp space

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Luzin N property

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Malliavin's absolute continuity lemma

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Measurable function

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Measurable space

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Measure (mathematics)

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Measure algebra

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Measure space

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Minkowski inequality

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Nikodym set

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Non-measurable set

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Null set

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Planar lamina

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Pointwise convergence

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Positive and negative sets

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Progressively measurable process

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Projection (measure theory)

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Radonifying function

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Rectifiable set

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Regular conditional probability

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Ruziewicz problem

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Set-theoretic limit

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Set function

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Sierpiński set

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Sigma-additive set function

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Simple function

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Smith–Volterra–Cantor set

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Solovay model

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Standard Borel space

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Standard probability space

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Strong measure zero set

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Sugeno integral

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Support (measure theory)

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Symmetric decreasing rearrangement

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Talagrand's concentration inequality

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Tightness of measures

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Vague topology

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Valuation (measure theory)

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Varifold

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Vitali covering lemma

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Vitali set

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Volterra's function

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Volume element

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Weierstrass function

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Τ-additivity

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Measure theory by

Ciro Santilli 35  Updated 2025-04-18  +Created 1970-01-01

Main motivation: Lebesgue integral.

The Bright Side Of Mathematics 2019 playlist: www.youtube.com/watch?v=xZ69KEg7ccU&list=PLBh2i93oe2qvMVqAzsX1Kuv6-4fjazZ8j

The key idea, is that we can't define a measure for the power set of R. Rather, we must select a large measurable subset, and the Borel sigma algebra is a good choice that matches intuitions.

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