Category: Theorems in measure theory

Monotone class theorem
In measure theory and probability, the monotone class theorem connects monotone classes and sigma-algebras. The theorem says that the smallest containing an algebra of sets is precisely the smallest 𝜎
Almgren regularity theorem
In geometric measure theory, a field of mathematics, the Almgren regularity theorem, proved by Almgren , states that the singular set of a mass-minimizing surface has codimension at least 2. Almgren's
Sard's theorem
In mathematics, Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis that asserts that the set of critical values (that is, the image of the set o
Vitali–Hahn–Saks theorem
In mathematics, the Vitali–Hahn–Saks theorem, introduced by Vitali, Hahn, and Saks, proves that under some conditions a sequence of measures converging point-wise does so uniformly and the limit is al
Fubini's theorem
In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. O
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 R of subsets of a given set Ω can be ext
Kakutani's theorem (measure theory)
In measure theory, a branch of mathematics, Kakutani's theorem is a fundamental result on the equivalence or mutual singularity of countable product measures. It gives an "if and only if" characterisa
F. and M. Riesz theorem
In mathematics, the F. and M. Riesz theorem is a result of the brothers Frigyes Riesz and Marcel Riesz, on analytic measures. It states that for a measure Ό on the circle, any part of Ό that is not ab
Regularity theorem for Lebesgue measure
No description available.
Steinhaus theorem
In the mathematical field of real analysis, the Steinhaus theorem states that the difference set of a set of positive measure contains an open neighbourhood of zero. It was first proved by Hugo Steinh
Lebesgue's decomposition theorem
In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem states that for every two σ-finite signed measures and on a measurable space there exist two σ-finite signed measures
CramĂ©r–Wold theorem
In mathematics, the CramĂ©r–Wold theorem in measure theory states that a Borel probability measure on is uniquely determined by the totality of its one-dimensional projections. It is used as a method f
Area formula (geometric measure theory)
In geometric measure theory the area formula relates the Hausdorff measure of the image of a Lipschitz map, while accounting for multiplicity, to the integral of the Jacobian of the map. It is one of
Prokhorov's theorem
In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematicia
Kuratowski and Ryll-Nardzewski measurable selection theorem
In mathematics, the Kuratowski–Ryll-Nardzewski measurable selection theorem is a result from measure theory that gives a sufficient condition for a multifunction to have a measurable selection functio
Fatou's lemma
In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma
Krylov–Bogolyubov theorem
In mathematics, the Krylov–Bogolyubov theorem (also known as the existence of invariant measures theorem) may refer to either of the two related fundamental theorems within the theory of dynamical sys
Lusin's theorem
In the mathematical field of real analysis, Lusin's theorem (or Luzin's theorem, named for Nikolai Luzin) or Lusin's criterion states that an almost-everywhere finite function is measurable if and onl
Hsu–Robbins–ErdƑs theorem
In the mathematical theory of probability, the Hsu–Robbins–ErdƑs theorem states that if is a sequence of i.i.d. random variables with zero mean and finite variance and then for every . The result was
Lebesgue differentiation theorem
In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages
Stein–Strömberg theorem
In mathematics, the Stein–Strömberg theorem or Stein–Strömberg inequality is a result in measure theory concerning the Hardy–Littlewood maximal operator. The result is foundational in the study of the
Bochner's theorem
In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's th
Vitali convergence theorem
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Hen
Ham sandwich theorem
In mathematical measure theory, for every positive integer n the ham sandwich theorem states that given n measurable "objects" in n-dimensional Euclidean space, it is possible to divide each one of th
Lebesgue's density theorem
In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set , the "density" of A is 0 or 1 at almost every point in . Additionally, the "density" of A is 1 at almost every p
Schröder–Bernstein theorem for measurable spaces
The Cantor–Bernstein–Schroeder theorem of set theory has a counterpart for measurable spaces, sometimes called the Borel Schroeder–Bernstein theorem, since measurable spaces are also called Borel spac
Monotone convergence theorem
In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or
Hahn decomposition theorem
In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that for any measurable space and any signed measure defined on the -algebra , there exist two
Clark–Ocone theorem
In mathematics, the Clark–Ocone theorem (also known as the Clark–Ocone–Haussmann theorem or formula) is a theorem of stochastic analysis. It expresses the value of some function F defined on the class
Bernstein's theorem on monotone functions
In real analysis, a branch of mathematics, Bernstein's theorem states that every real-valued function on the half-line [0, ∞) that is totally monotone is a mixture of exponential functions. In one imp
Doob–Dynkin lemma
In probability theory, the Doob–Dynkin lemma, named after Joseph L. Doob and Eugene Dynkin, characterizes the situation when one random variable is a function of another by the inclusion of the -algeb
Fatou–Lebesgue theorem
In mathematics, the Fatou–Lebesgue theorem establishes a chain of inequalities relating the integrals (in the sense of Lebesgue) of the limit inferior and the limit superior of a sequence of functions
Fernique's theorem
In mathematics - specifically, in measure theory - Fernique's theorem is a result about Gaussian measures on Banach spaces. It extends the finite-dimensional result that a Gaussian random variable has
Scheffé's lemma
In mathematics, Scheffé's lemma is a proposition in measure theory concerning the convergence of sequences of integrable functions. It states that, if is a sequence of integrable functions on a measur
Alexandrov theorem
In mathematical analysis, the Alexandrov theorem, named after Aleksandr Danilovich Aleksandrov, states that if U is an open subset of and is a convex function, then has a second derivative almost ever
Egorov's theorem
In measure theory, an area of mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions. It is also named Severini–Eg
Kƍmura's theorem
In mathematics, Kƍmura's theorem is a result on the differentiability of absolutely continuous Banach space-valued functions, and is a substantial generalization of Lebesgue's theorem on the different
Fubini's theorem on differentiation
In mathematics, Fubini's theorem on differentiation, named after Guido Fubini, is a result in real analysis concerning the differentiation of series of monotonic functions. It can be proven by using F
Naimark's dilation theorem
In operator theory, Naimark's dilation theorem is a result that characterizes positive operator valued measures. It can be viewed as a consequence of Stinespring's dilation theorem.
Feldman–Hájek theorem
In probability theory, the Feldman–Hájek theorem or Feldman–Hájek dichotomy is a fundamental result in the theory of Gaussian measures. It states that two Gaussian measures and on a locally convex spa
Hobby–Rice theorem
In mathematics, and in particular the necklace splitting problem, the Hobby–Rice theorem is a result that is useful in establishing the existence of certain solutions. It was proved in 1965 by Charles
Borel–Cantelli lemma
In probability theory, the Borel–Cantelli lemma is a theorem about sequences of events. In general, it is a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli, who ga
Radon–Nikodym theorem
In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A measure is a set function that assi
Disintegration theorem
In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of
Maharam's theorem
In mathematics, Maharam's theorem is a deep result about the decomposability of measure spaces, which plays an important role in the theory of Banach spaces. In brief, it states that every complete me
Parthasarathy's theorem
In mathematics – and in particular the study of games on the unit square – Parthasarathy's theorem is a generalization of Von Neumann's minimax theorem. It states that a particular class of games has
Dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the L1 norm. Its
Stromquist–Woodall theorem
The Stromquist–Woodall theorem is a theorem in fair division and measure theory. Informally, it says that, for any cake, for any n people with different tastes, and for any fraction r, there exists a
Stahl's theorem
In matrix analysis Stahl's theorem is a theorem proved in 2011 by Herbert Stahl concerning Laplace transforms for special matrix functions. It originated in 1975 as the Bessis-Moussa-Villani (BMV) con
Brunn–Minkowski theorem
In mathematics, the Brunn–Minkowski theorem (or Brunn–Minkowski inequality) is an inequality relating the volumes (or more generally Lebesgue measures) of compact subsets of Euclidean space. The origi
Dubins–Spanier theorems
The Dubins–Spanier theorems are several theorems in the theory of fair cake-cutting. They were published by Lester Dubins and Edwin Spanier in 1961. Although the original motivation for these theorems
Rademacher's theorem
In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If U is an open subset of Rn and f: U → Rm is Lipschitz continuous, then f is differentiable almost e
Cameron–Martin theorem
In mathematics, the Cameron–Martin theorem or Cameron–Martin formula (named after Robert Horton Cameron and W. T. Martin) is a theorem of measure theory that describes how abstract Wiener measure chan
Structure theorem for Gaussian measures
In mathematics, the structure theorem for Gaussian measures shows that the abstract Wiener space construction is essentially the only way to obtain a strictly positive Gaussian measure on a separable