Linked from

The 35 pages that link to Measure theory, each with the reason it gives.

• Probability theoryNarrower topic: Kolmogorov’s formal framework builds probability on measure-theoretic foundations.

• Probability measureNarrower topic: Probability measures are measures specialized to have total mass one.

• CalculusNarrower topic: Lebesgue integration generalizes the integral beyond the basic constructions of introductory calculus.

• CardinalityCompared with: Measure captures geometric size that cardinality cannot distinguish between most subsets of the real line.

• Andrey KolmogorovRelated: Its language let Kolmogorov define probability rigorously on general event spaces.

• Borel–Cantelli lemmaNarrower topic: The lemmas express probability as a measure on event sequences.

• Functional analysisRelated: Measure supplies the integrals used to build many function spaces.

• IntegralNarrower topic: Lebesgue integration rests on its account of size and measurable functions.

• Émile BorelNarrower topic: It is the field in which Borel’s work on sets and functions became foundational.

• Poincaré recurrence theoremNarrower topic: The theorem's assumptions and its almost-everywhere conclusion are expressed in measure-theoretic terms.

• Banach–Tarski paradoxNarrower topic: The theorem marks a boundary on extending volume consistently to all sets.

• Ergodic theoryNarrower topic: Its definitions make probability and invariant averages precise.

• Henri LebesgueNarrower topic: Lebesgue built this framework to generalize length and underpin his integral.

• Integral calculusNarrower topic: Its framework extends integration beyond the intervals and functions of elementary calculus.

• History of calculusRelated: Its development broadened integration beyond the classical Riemann framework.

• Cavalieri's principleNarrower topic: It clarifies which cross-sections and figures have well-defined measurable size.

• Multiple integralNarrower topic: It supplies the general notion of size underlying integration over complex domains.

• Prokhorov's theoremNarrower topic: Prokhorov’s result builds on the measure-theoretic formulation of probability.

• Constantin CarathéodoryRelated: Carathéodory developed an outer-measure construction that became a foundation for modern measure theory.

• Integration by substitutionNarrower topic: The substitution rule extends to integration with respect to measures through broader transformation theorems.

• Multivariable calculusRelated: It supplies a broader foundation for integration beyond elementary regions.

• Probability axiomsNarrower topic: Kolmogorov modeled probability as a normalized measure.

• Volume integralNarrower topic: It formalizes integration over general regions and functions.

• Carathéodory's existence theoremNarrower topic: Lebesgue measurability and integrability let the theorem handle right-hand sides beyond continuous functions.

• Crofton formulaRelated: A precise average over infinitely many lines depends on measure and integration.

• Fatou's lemmaNarrower topic: Fatou's lemma is a foundational theorem in measure-theoretic integration.

• Hausdorff paradoxNarrower topic: The decomposition exposed limits on extending ordinary area to all subsets of a sphere.

• History of probabilityRelated: It supplied the language needed to define probability on continuous and abstract spaces.

• Poisson point processNarrower topic: Intensity measures use its language to assign expected counts to arbitrary measurable regions.

• Young's inequality for productsRelated: Integral forms of Young-type estimates rely on the measure-theoretic framework for integration.

• Fractal analysisNarrower topic: Dimension definitions rely on precise ways to measure sets and coverings.

• Pizza theoremNarrower topic: Measure theory supplies the language for extending area identities to general measurable regions.

• Boole's inequalityRelated: Countable subadditivity extends the inequality from events to measurable sets.

• Infinite monkey theoremNarrower topic: The theorem's rigorous probability language grew from measure-theoretic foundations.

• Lonely runner conjectureNarrower topic: Measure bounds can show that forbidden time intervals cannot cover every possible time.