Lebesgue measure
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Lebesgue measure — In mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration. Sets… … Wikipedia
Regularity theorem for Lebesgue measure — In mathematics, the regularity theorem for Lebesgue measure is a result in measure theory that states that Lebesgue measure on the real line is a regular measure. Informally speaking, this means that every Lebesgue measurable subset of the real… … Wikipedia
There is no infinite-dimensional Lebesgue measure — In mathematics, it is a theorem that there is no analogue of Lebesgue measure on an infinite dimensional space. This fact forces mathematicians studying measure theory on infinite dimensional spaces to use other kinds of measures: often, the… … Wikipedia
Lebesgue integration — In mathematics, the integral of a non negative function can be regarded in the simplest case as the area between the graph of that function and the x axis. Lebesgue integration is a mathematical construction that extends the integral to a larger… … Wikipedia
Measure (mathematics) — Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. In mathematical analysis … Wikipedia
Lebesgue-Stieltjes integration — In measure theoretic analysis and related branches of mathematics, Lebesgue Stieltjes integration generalizes Riemann Stieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure theoretic framework.… … Wikipedia
Lebesgue's density theorem — In mathematics, Lebesgue s density theorem states that for any Lebesgue measurable set A, the density of A is 1 at almost every point in A. Intuitively, this means that the edge of A, the set of points in A whose neighborhood is partially in A… … Wikipedia
measure theory — noun A branch of mathematical analysis, concerned with the theory of integration, that generalizes the intuitive notions of length, area and volume. See Also: Borel measure, complex measure, Haar measur … Wiktionary
Lebesgue point — In mathematics, given a Lebesgue integrable function f, a point x in the domain of f is a Lebesgue point if:lim {r ightarrow 0^+}frac{1}int {B(x,r)} !|f(y) f(x)|,dy=0.Here, B(x,r) is the ball centered at x with radius r, and |B(x,r)| is the… … Wikipedia
measure — measurer, n. /mezh euhr/, n., v., measured, measuring. n. 1. a unit or standard of measurement: weights and measures. 2. a system of measurement: liquid measure. 3. an instrument, as a graduated rod or a container of standard capacity, for… … Universalium
