Lebesgue measure

Lebesgue measure assigns a size to subsets of Euclidean space, extending length, area, and volume while accommodating many irregular sets.

Lebesgue outer measure: A set function defined by covering subsets of Euclidean space with countably many boxes and taking the least total volume. It supplies the covering-based size from which Lebesgue measure is constructed.
Interval: A connected subset of the real line containing every point between any two of its points. Lengths of intervals anchor Lebesgue measure on the real line.
Lebesgue integral: An integral defined by measuring the sizes of a function's level sets rather than partitioning its domain into intervals. Lebesgue measure provides the sizes used to define this integral.
Henri Lebesgue: A French mathematician whose work established the theory of measure and the Lebesgue integral. His early twentieth-century work developed the measure named after him.
Measurable set: A set satisfying Carathéodory’s criterion, which makes its outer measure split consistently across every test set. This criterion identifies which sets receive a well-behaved Lebesgue measure.
Borel set: A member of the smallest sigma-algebra containing every open subset of a topological space. Borel sets form a basic domain on which Lebesgue measure is first defined.
Lebesgue differentiation theorem: A theorem stating that local averages of an integrable function converge to its value almost everywhere. Its almost-everywhere conclusion is expressed using Lebesgue measure.
Émile Borel: A French mathematician who developed the theory of Borel sets and contributed to measure theory. Borel's sets provided a key foundation for Lebesgue's construction.
Carathéodory's criterion: A condition requiring a set to split every test set's outer measure additively into its parts inside and outside. It characterizes the measurable sets used in Lebesgue measure's construction.
Sigma-algebra: A collection of subsets closed under complements and countable unions. Lebesgue measure is defined on a sigma-algebra of subsets.

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