Table of Contents
86 relations: Algebraic number, Ambient space (mathematics), Annals of Mathematics, Area, Argumentum a fortiori, Axiom of choice, Axiom of determinacy, Banach–Tarski paradox, Borel measure, Borel set, Cantor set, Carathéodory's criterion, Carathéodory's extension theorem, Cartesian product, Closed set, Complement (set theory), Complete measure, Countable set, Curve, Dense set, Dimension, Disjoint union, Dragon curve, Edison Farah, Euclidean distance, Euclidean space, Fσ set, Four-dimensional space, Fractal, France, Gδ set, Haar measure, Hausdorff dimension, Hausdorff measure, Henri Lebesgue, Hyperplane, Infimum and supremum, Infinite-dimensional Lebesgue measure, Inner regular measure, Intersection (set theory), Interval (mathematics), Σ-algebra, Σ-finite measure, Lebesgue covering dimension, Lebesgue integral, Lebesgue's density theorem, Length, Linear map, Liouville number, Locally compact group, ... Expand index (36 more) »
Algebraic number
An algebraic number is a number that is a root of a non-zero polynomial (of finite degree) in one variable with integer (or, equivalently, rational) coefficients.
See Lebesgue measure and Algebraic number
Ambient space (mathematics)
In mathematics, especially in geometry and topology, an ambient space is the space surrounding a mathematical object along with the object itself.
See Lebesgue measure and Ambient space (mathematics)
Annals of Mathematics
The Annals of Mathematics is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
See Lebesgue measure and Annals of Mathematics
Area
Area is the measure of a region's size on a surface.
Argumentum a fortiori
Argumentum a fortiori (literally "argument from the stronger ") is a form of argumentation that draws upon existing confidence in a proposition to argue in favor of a second proposition that is held to be implicit in, and even more certain than, the first.
See Lebesgue measure and Argumentum a fortiori
Axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty.
See Lebesgue measure and Axiom of choice
Axiom of determinacy
In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962.
See Lebesgue measure and Axiom of determinacy
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.
See Lebesgue measure and Banach–Tarski paradox
Borel measure
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Lebesgue measure and Borel measure are measures (measure theory).
See Lebesgue measure and Borel measure
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement.
See Lebesgue measure and Borel set
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties.
See Lebesgue measure and Cantor set
Carathéodory's criterion
Carathéodory's criterion is a result in measure theory that was formulated by Greek mathematician Constantin Carathéodory that characterizes when a set is Lebesgue measurable.
See Lebesgue measure and Carathéodory's criterion
Carathéodory's extension theorem
In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given ring of subsets R of a given set Ω can be extended to a measure on the σ-ring generated by R, and this extension is unique if the pre-measure is σ-finite.
See Lebesgue measure and Carathéodory's extension theorem
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and, denoted, is the set of all ordered pairs where is in and is in.
See Lebesgue measure and Cartesian product
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.
See Lebesgue measure and Closed set
Complement (set theory)
In set theory, the complement of a set, often denoted by A^\complement, is the set of elements not in.
See Lebesgue measure and Complement (set theory)
Complete measure
In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). Lebesgue measure and complete measure are measures (measure theory).
See Lebesgue measure and Complete measure
Countable set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers.
See Lebesgue measure and Countable set
Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
See Lebesgue measure and Curve
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).
See Lebesgue measure and Dense set
Dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.
See Lebesgue measure and Dimension
Disjoint union
In mathematics, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come.
See Lebesgue measure and Disjoint union
Dragon curve
A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems.
See Lebesgue measure and Dragon curve
Edison Farah
Edison Farah (Capivari, April 14, 1915 - São Paulo, April 14, 2006) was a Brazilian mathematician, professor at the University of São Paulo.
See Lebesgue measure and Edison Farah
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them.
See Lebesgue measure and Euclidean distance
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space.
See Lebesgue measure and Euclidean space
Fσ set
In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets.
See Lebesgue measure and Fσ set
Four-dimensional space
Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D).
See Lebesgue measure and Four-dimensional space
Fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension.
See Lebesgue measure and Fractal
France
France, officially the French Republic, is a country located primarily in Western Europe.
See Lebesgue measure and France
Gδ set
In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets.
See Lebesgue measure and Gδ set
Haar measure
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. Lebesgue measure and Haar measure are measures (measure theory).
See Lebesgue measure and Haar measure
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff.
See Lebesgue measure and Hausdorff dimension
Hausdorff measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. Lebesgue measure and Hausdorff measure are measures (measure theory).
See Lebesgue measure and Hausdorff measure
Henri Lebesgue
Henri Léon Lebesgue (June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of a function defined for that axis.
See Lebesgue measure and Henri Lebesgue
Hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension.
See Lebesgue measure and Hyperplane
Infimum and supremum
In mathematics, the infimum (abbreviated inf;: infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists.
See Lebesgue measure and Infimum and supremum
Infinite-dimensional Lebesgue measure
An infinite-dimensional Lebesgue measure is a type of measure defined on an infinite-dimensional normed vector space, more specifically a Banach space.
See Lebesgue measure and Infinite-dimensional Lebesgue measure
Inner regular measure
In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets. Lebesgue measure and inner regular measure are measures (measure theory).
See Lebesgue measure and Inner regular measure
Intersection (set theory)
In set theory, the intersection of two sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A.
See Lebesgue measure and Intersection (set theory)
Interval (mathematics)
In mathematics, a (real) interval is the set of all real numbers lying between two fixed endpoints with no "gaps".
See Lebesgue measure and Interval (mathematics)
Σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections.
See Lebesgue measure and Σ-algebra
Σ-finite measure
In mathematics, a positive (or signed) measure μ defined on a ''σ''-algebra Σ of subsets of a set X is called a finite measure if μ(X) is a finite real number (rather than ∞). Lebesgue measure and Σ-finite measure are measures (measure theory).
See Lebesgue measure and Σ-finite measure
Lebesgue covering dimension
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way.
See Lebesgue measure and Lebesgue covering dimension
Lebesgue integral
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the axis.
See Lebesgue measure and Lebesgue integral
Lebesgue's density theorem
In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set A\subset \R^n, the "density" of A is 0 or 1 at almost every point in \R^n.
See Lebesgue measure and Lebesgue's density theorem
Length
Length is a measure of distance.
See Lebesgue measure and Length
Linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication.
See Lebesgue measure and Linear map
Liouville number
In number theory, a Liouville number is a real number x with the property that, for every positive integer n, there exists a pair of integers (p,q) with q>1 such that The inequality implies that Liouville numbers possess an excellent sequence of rational number approximations.
See Lebesgue measure and Liouville number
Locally compact group
In mathematics, a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff.
See Lebesgue measure and Locally compact group
Locally finite measure
In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure. Lebesgue measure and locally finite measure are measures (measure theory).
See Lebesgue measure and Locally finite measure
Mathematics
Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.
See Lebesgue measure and Mathematics
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. Lebesgue measure and measure (mathematics) are measures (measure theory).
See Lebesgue measure and Measure (mathematics)
Metric space
In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points.
See Lebesgue measure and Metric space
Non-measurable set
In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "volume".
See Lebesgue measure and Non-measurable set
Null set
In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero.
See Lebesgue measure and Null set
Open set
In mathematics, an open set is a generalization of an open interval in the real line.
See Lebesgue measure and Open set
Osgood curve
In mathematical analysis, an Osgood curve is a non-self-intersecting curve that has positive area.
See Lebesgue measure and Osgood curve
Outer measure
In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. Lebesgue measure and outer measure are measures (measure theory).
See Lebesgue measure and Outer measure
Peano curve
In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890.
See Lebesgue measure and Peano curve
Peano–Jordan measure
In mathematics, the Peano–Jordan measure (also known as the Jordan content) is an extension of the notion of size (length, area, volume) to shapes more complicated than, for example, a triangle, disk, or parallelepiped. Lebesgue measure and Peano–Jordan measure are measures (measure theory).
See Lebesgue measure and Peano–Jordan measure
Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. Lebesgue measure and Radon measure are measures (measure theory).
See Lebesgue measure and Radon measure
Real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions.
See Lebesgue measure and Real analysis
Real number
In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.
See Lebesgue measure and Real number
Rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles.
See Lebesgue measure and Rectangle
Rectangular cuboid
A rectangular cuboid is a special case of a cuboid with rectangular faces in which all of its dihedral angles are right angles.
See Lebesgue measure and Rectangular cuboid
Robert M. Solovay
Robert Martin Solovay (born December 15, 1938) is an American mathematician working in set theory.
See Lebesgue measure and Robert M. Solovay
Rudolf Lipschitz
Rudolf Otto Sigismund Lipschitz (14 May 1832 – 7 October 1903) was a German mathematician who made contributions to mathematical analysis (where he gave his name to the Lipschitz continuity condition) and differential geometry, as well as number theory, algebras with involution and classical mechanics.
See Lebesgue measure and Rudolf Lipschitz
Series (mathematics)
In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity.
See Lebesgue measure and Series (mathematics)
Set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.
See Lebesgue measure and Set theory
Sign (mathematics)
In mathematics, the sign of a real number is its property of being either positive, negative, or 0.
See Lebesgue measure and Sign (mathematics)
Smith–Volterra–Cantor set
In mathematics, the Smith–Volterra–Cantor set (SVC), ε-Cantor set, or fat Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), yet has positive measure.
See Lebesgue measure and Smith–Volterra–Cantor set
Solovay model
In the mathematical field of set theory, the Solovay model is a model constructed by in which all of the axioms of Zermelo–Fraenkel set theory (ZF) hold, exclusive of the axiom of choice, but in which all sets of real numbers are Lebesgue measurable.
See Lebesgue measure and Solovay model
Strictly positive measure
In mathematics, strict positivity is a concept in measure theory. Lebesgue measure and Strictly positive measure are measures (measure theory).
See Lebesgue measure and Strictly positive measure
Submanifold
In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S \rightarrow M satisfies certain properties.
See Lebesgue measure and Submanifold
Subset
In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment).
See Lebesgue measure and Subset
Support (measure theory)
In mathematics, the support (sometimes topological support or spectrum) of a measure \mu on a measurable topological space (X, \operatorname(X)) is a precise notion of where in the space X the measure "lives". Lebesgue measure and support (measure theory) are measures (measure theory).
See Lebesgue measure and Support (measure theory)
Symmetric difference
In mathematics, the symmetric difference of two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection.
See Lebesgue measure and Symmetric difference
Synonym
A synonym is a word, morpheme, or phrase that means precisely or nearly the same as another word, morpheme, or phrase in a given language.
See Lebesgue measure and Synonym
Translational symmetry
In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation (without rotation).
See Lebesgue measure and Translational symmetry
Uncountable set
In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable.
See Lebesgue measure and Uncountable set
Union (set theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.
See Lebesgue measure and Union (set theory)
Vitali set
In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905.
See Lebesgue measure and Vitali set
Volume
Volume is a measure of regions in three-dimensional space.
See Lebesgue measure and Volume
Volume of an n-ball
In geometry, a ball is a region in a space comprising all points within a fixed distance, called the radius, from a given point; that is, it is the region enclosed by a sphere or hypersphere.
See Lebesgue measure and Volume of an n-ball
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox.
See Lebesgue measure and Zermelo–Fraenkel set theory
References
Also known as Hyper-volume, Hyper-volumes, Hypervolume, Lebesgue measurable, Lebesgue measurable set, Lebesgue null, Lebesgue outer measure, Lebesgue-measurable, Lebesgue-measurable set, Lesbegue measure, N-volume, Regularity theorem for Lebesgue measure, Zero Lebesgue measure.