Henri Lebesgue

Henri Lebesgue (1875–1941) was a French mathematician who developed measure theory and the Lebesgue integral, reshaping modern analysis.

Measure theory: The mathematical study of measures, which assign sizes to sets while supporting operations such as integration. Lebesgue built this framework to generalize length and underpin his integral.
Simple function: A measurable function taking only finitely many values. Lebesgue integration begins by assigning integrals to these finite-valued approximations.
Borel measure: A measure defined on the sigma-algebra generated by open sets in a topological space. Émile Borel’s work on sets and measure helped establish the setting Lebesgue extended.
Lp space: A space of measurable functions whose absolute values raised to a specified power have finite integrals. Lebesgue integration defines the norms that make these function spaces possible.
Lebesgue measure: A measure on Euclidean spaces that extends ordinary length, area, and volume to broad classes of sets. It supplied the notion of size behind Lebesgue’s integration theory.
Monotone convergence theorem: A theorem stating that integrals of an increasing sequence of nonnegative measurable functions converge to the integral of their limit. It lets the integral pass from simple approximations to increasing limits.
Émile Borel: A French mathematician whose work in measure theory, probability, and analysis influenced Lebesgue. Borel’s theories of sets and measure formed part of the intellectual background to Lebesgue’s work.
Lebesgue–Stieltjes integral: An integral that combines Lebesgue integration with integration against a measure induced by a function. It extends Lebesgue’s method to measures generated by monotone functions.
Measurable set: A set to which a specified measure can consistently assign a size. Measurable sets are the domain on which Lebesgue measure and integration operate.
Dominated convergence theorem: A theorem allowing limits to pass through integrals when functions are bounded in absolute value by one integrable function. This central limit theorem showcases the strength of Lebesgue’s integration framework.

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