Lebesguemåttets

Lebesgue measure is a measure defined on subsets of n-dimensional Euclidean space that extends the usual notions of length, area, and volume. It is the canonical translation-invariant measure used in real analysis and probability. The measure is named after Henri Lebesgue and serves as the standard reference measure for integration on R^n.

Construction and definition: For a subset E of R^n, the Lebesgue outer measure m*(E) is defined as the infimum of the total volumes of countable coverings of E by rectangles. A set is Lebesgue measurable if, for every A ⊆ R^n, m*(A) = m*(A ∩ E) + m*(A \ E) (the Carathéodory criterion). The Lebesgue measure m is then defined on the collection of measurable sets, and it is complete, meaning all subsets of measure-zero sets are measurable with measure zero. It agrees with the ordinary volume on simple sets and extends the Borel measure.

Key properties: Lebesgue measure is translation invariant: m(E + x) = m(E) for all x ∈ R^n. For a linear map L, m(L(E)) = |det L| m(E). The measure is sigma-finite on R^n and is regular, with outer regularity by open sets and inner regularity by compact sets in many contexts.

Applications: Lebesgue measure underpins the Lebesgue integral, enabling robust convergence theorems such as monotone and dominated convergence, as well as Fubini’s theorem. It is foundational in modern analysis, probability theory, harmonic analysis, and partial differential equations.

Generalizations: The concept extends to product spaces and more abstract measure spaces, and relates to Haar measure, the unique translation-invariant measure on locally compact groups.