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79 relations: Annals of Mathematics, Area, Argumentum a fortiori, Axiom, Axiom of choice, Axiom of determinacy, Banach–Tarski paradox, Borel measure, Borel set, Cantor set, Carathéodory's extension theorem, Cartesian product, Closed set, Complement (set theory), Complete measure, Countable set, Curve, Dense set, Disjoint union, Dragon curve, Euclidean distance, Euclidean space, Fσ set, Fractal, France, Gδ set, Haar measure, Hausdorff dimension, Hausdorff measure, Henri Lebesgue, Infimum and supremum, Infinite-dimensional Lebesgue measure, Inner regular measure, Intersection (set theory), Interval (mathematics), Σ-finite measure, Lebesgue covering dimension, Lebesgue integration, Lebesgue's density theorem, Length, Linear map, Locally compact space, Locally finite measure, Measure (mathematics), Metric space, Non-measurable set, Null set, Open set, Outer measure, Peano curve, ..., Power set, Radon measure, Rational number, Real analysis, Real number, Rectangle, Regularity theorem for Lebesgue measure, Robert M. Solovay, Rudolf Lipschitz, Series (mathematics), Set theory, Sigma-algebra, Sign (mathematics), Smith–Volterra–Cantor set, Solovay model, Strictly positive measure, Submanifold, Subset, Support (measure theory), Symmetric difference, Synonym, Topological group, Translational symmetry, Uncountable set, Union (set theory), Vitali set, Volume, Volume form, Zermelo–Fraenkel set theory. Expand index (29 more) »
Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.
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Area
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane.
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Argumentum a fortiori
Argumentum a fortiori (Latin: "from a/the stronger ") is a form of argumentation which draws upon existing confidence in a proposition to argue in favor of a second proposition that is held to be implicit in the first.
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Axiom
An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.
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Axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.
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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.
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Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑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.
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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).
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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.
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Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties.
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Carathéodory's extension theorem
In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any measure defined on a given ring R of subsets of a given set Ω can be extended to the σ-algebra generated by R, and this extension is unique if the measure is σ-finite.
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Cartesian product
In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.
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Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.
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Complement (set theory)
In set theory, the complement of a set refers to elements not in.
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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).
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Countable set
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.
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Curve
In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.
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Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A, that is the closure of A is constituting the whole set X. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A — for instance, every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).
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Disjoint union
In set theory, the disjoint union (or discriminated union) of a family of sets is a modified union operation that indexes the elements according to which set they originated in.
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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.
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Euclidean distance
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space.
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Euclidean space
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
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Fσ set
In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets.
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Fractal
In mathematics, a fractal is an abstract object used to describe and simulate naturally occurring objects.
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France
France, officially the French Republic (République française), is a sovereign state whose territory consists of metropolitan France in Western Europe, as well as several overseas regions and territories.
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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.
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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.
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Hausdorff dimension
Hausdorff dimension is a measure of roughness in mathematics introduced in 1918 by mathematician Felix Hausdorff, and it serves as a measure of the local size of a space, taking into account the distance between its points.
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Hausdorff measure
In mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in to each set in Rn or, more generally, in any metric space.
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Henri Lebesgue
Henri Léon Lebesgue (June 28, 1875 – July 26, 1941) was a French mathematician most famous 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.
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Infimum and supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.
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Infinite-dimensional Lebesgue measure
In mathematics, it is a theorem that there is no analogue of Lebesgue measure on an infinite-dimensional Banach space.
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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.
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Intersection (set theory)
In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.
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Interval (mathematics)
In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.
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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 finite if μ(X) is a finite real number (rather than ∞).
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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.
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Lebesgue integration
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.
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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.
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Length
In geometric measurements, length is the most extended dimension of an object.
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Linear map
In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.
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Locally compact space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.
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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.
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Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.
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Metric space
In mathematics, a metric space is a set for which distances between all members of the set are defined.
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Non-measurable set
In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "size".
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Null set
In set theory, a null set N \subset \mathbb is a set that can be covered by a countable union of intervals of arbitrarily small total length.
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Open set
In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.
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Outer measure
In mathematics, in particular in 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.
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Peano curve
In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890.
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Power set
In mathematics, the power set (or powerset) of any set is the set of all subsets of, including the empty set and itself, variously denoted as, 𝒫(), ℘() (using the "Weierstrass p"),,, or, identifying the powerset of with the set of all functions from to a given set of two elements,.
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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 X that is locally finite and inner regular.
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Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
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Real analysis
In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles.
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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.
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Robert M. Solovay
Robert Martin Solovay (born December 15, 1938) is an American mathematician specializing in set theory.
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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.
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Series (mathematics)
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.
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Set theory
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.
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Sigma-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that includes the empty subset, is closed under complement, and is closed under countable unions and countable intersections.
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Sign (mathematics)
In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative.
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Smith–Volterra–Cantor set
In mathematics, the Smith–Volterra–Cantor set (SVC), fat Cantor set, or ε-Cantor set is an example of a set of points on the real line ℝ that is nowhere dense (in particular it contains no intervals), yet has positive measure.
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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.
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Strictly positive measure
In mathematics, strict positivity is a concept in measure theory.
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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 → M satisfies certain properties.
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Subset
In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.
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Support (measure theory)
In mathematics, the support (sometimes topological support or spectrum) of a measure μ on a measurable topological space (X, Borel(X)) is a precise notion of where in the space X the measure "lives".
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Symmetric difference
In mathematics, the symmetric difference, also known as the disjunctive union, of two sets is the set of elements which are in either of the sets and not in their intersection.
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Synonym
A synonym is a word or phrase that means exactly or nearly the same as another word or phrase in the same language.
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Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.
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Translational symmetry
In geometry, a translation "slides" a thing by a: Ta(p).
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Uncountable set
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable.
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Union (set theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.
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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.
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Volume
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.
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Volume form
In mathematics, a volume form on a differentiable manifold is a top-dimensional form (i.e., a differential form of top degree).
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Zermelo–Fraenkel set theory
In mathematics, 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.
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Redirects here:
Lebesgue measurable, Lebesgue measurable set, Lebesgue null, Lebesgue sigma-algebra, Lebesgue-measurable, Lesbegue measure, Zero Lebesgue measure.
References
[1] https://en.wikipedia.org/wiki/Lebesgue_measure
