Linked from

The 43 pages that link to Lebesgue measure, each with the reason it gives.

• Lebesgue integralRelated: On the real line, it supplies the sizes used in the standard Lebesgue integral.

• AreaNarrower topic: It formalizes area for sets more complicated than ordinary geometric figures.

• Measure theoryBroader topic: It is the standard model for measuring subsets of Euclidean space.

• Sigma-algebraRelated: Its domain is the Lebesgue sigma-algebra, which includes all Borel sets and their completion.

• Probability spaceCompared with: It need not have total measure one, unlike the measure of a probability space.

• Almost everywhereRelated: It supplies the standard meaning of almost everywhere in real analysis.

• Cantor setRelated: The lengths removed sum to the entire original interval, so the Cantor set has measure zero.

• Countable additivityBroader topic: Its countable additivity supports measuring complicated unions beyond intervals and rectangles.

• Émile BorelRelated: Borel’s work belongs to the measure-theoretic foundations that Lebesgue measure helped establish.

• Hausdorff dimensionCompared with: It captures ordinary volume, while Hausdorff measure adapts its exponent to lower-dimensional sets.

• Lebesgue differentiation theoremNarrower topic: The theorem measures exceptional points and normalizes ball averages using Lebesgue measure.

• Measure spaceBroader topic: It assigns familiar geometric sizes within a measure-space framework.

• Banach–Tarski paradoxRelated: Its finite additivity explains why measurable pieces could not double a ball's volume.

• Borel measureCompared with: Its domain includes all Lebesgue-measurable sets, extending beyond the Borel sets.

• Geometric measure theoryNarrower topic: It supplies the measure-theoretic framework used to integrate over geometric sets.

• Henri LebesgueBroader topic: It supplied the notion of size behind Lebesgue’s integration theory.

• Uncountable setRelated: Uncountability does not imply positive measure, as the Cantor set demonstrates.

• Haar measureCompared with: It is the familiar additive-group instance of Haar measure.

• Absolute continuityNarrower topic: The definition measures the combined lengths of the intervals where changes are tested.

• Hardy–Littlewood maximal functionNarrower topic: It measures both the averaging regions and the maximal function’s level sets.

• Nonmeasurable setBroader topic: Its domain includes every Borel set and every subset of a set of measure zero.

• Vitali covering theoremNarrower topic: The classical theorem identifies the uncovered remainder as Lebesgue-null.

• Constantin CarathéodoryRelated: Carathéodory's outer-measure method provides a standard construction of Lebesgue measure.

• Ham sandwich theoremRelated: It models ordinary geometric volume when objects are treated as regions.

• Carathéodory's extension theoremBroader topic: Its construction extends interval lengths from a simple algebra to a σ-algebra of measurable sets.

• Lebesgue decomposition theoremRelated: Decomposition relative to Lebesgue measure separates densities from singular distributions.

• Lebesgue density theoremNarrower topic: The theorem's proportions and exceptional sets are defined using this measure.

• Volume integralNarrower topic: It provides the measure-theoretic foundation for integration over regions.

• Besicovitch covering theoremRelated: Overlap bounds convert sums of ball volumes into estimates of covered sets.

• Hausdorff paradoxCompared with: Its additivity and rotation invariance cannot extend to every piece in the paradoxical partition.

• Solovay modelNarrower topic: Measurability in the model is defined relative to this standard measure.

• Vitali covering lemmaRelated: The lemma’s covering conclusions frequently support estimates for Lebesgue measure.

• Cardinality of the continuumRelated: It distinguishes the continuum-sized interval from its measure-zero subsets, such as the rationals.

• Continuous uniform distributionRelated: On an interval, uniform probability is normalized length.

• Rademacher's theoremRelated: It specifies what ‘almost everywhere’ means in the theorem.

• Sard's theoremRelated: Measure zero gives the theorem's conclusion its rigorous meaning.

• Steinhaus theoremNarrower topic: Positive finite measure supplies the nonzero overlap from which the neighborhood follows.

• Borel conjectureRelated: Strong measure zero is a covering property distinct from having Lebesgue measure zero.

• Brunn–Minkowski theoremRelated: It supplies the volume notion used in the measurable-set formulation.

• Denjoy–Luzin theoremNarrower topic: The theorem measures the exceptional discontinuity set using Lebesgue measure.

• F. and M. Riesz theoremRelated: Arc length on the circle is the reference measure in the theorem's conclusion.

• Fuglede's conjectureRelated: Measurability and almost-everywhere tiling make the conjecture measure-theoretic.

• Lochs's theoremRelated: It specifies the sense in which the theorem's exceptional real numbers have measure zero.