|
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" characterisation of when two such measures are equivalent, and hence it is extremely useful when trying to establish change-of-measure formulae for measures on function spaces. The result is due to the Japanese mathematician Shizuo Kakutani. Kakutani's theorem can be used, for example, to determine whether a translate of a Gaussian measure In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are nam ... \mu is equivalent to \mu (only when the translation vector lies in the Cameron–Martin space of \mu), or whether a dilation of \mu is equivalent to \mu (only when the absolute value of the dilation factor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure Theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence (measure Theory)
In mathematics, and specifically in measure theory, equivalence is a notion of two measures being qualitatively similar. Specifically, the two measures agree on which events have measure zero. Definition Let \mu and \nu be two measures on the measurable space (X, \mathcal A), and let :\mathcal_\mu := \ and :\mathcal_\nu := \ be the sets of \mu-null sets and \nu-null sets, respectively. Then the measure \nu is said to be absolutely continuous in reference to \mu iff \mathcal N_\nu \supseteq \mathcal N_\mu. This is denoted as \nu \ll \mu. The two measures are called equivalent iff \mu \ll \nu and \nu \ll \mu, which is denoted as \mu \sim \nu. That is, two measures are equivalent if they satisfy \mathcal N_\mu = \mathcal N_\nu. Examples On the real line Define the two measures on the real line as : \mu(A)= \int_A \mathbf 1_(x) \mathrm dx : \nu(A)= \int_A x^2 \mathbf 1_(x) \mathrm dx for all Borel sets A . Then \mu and \nu are equivalent, since all sets outside of ,1 have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Measure
In mathematics, two positive (or signed or complex) measures \mu and \nu defined on a measurable space (\Omega, \Sigma) are called singular if there exist two disjoint measurable sets A, B \in \Sigma whose union is \Omega such that \mu is zero on all measurable subsets of B while \nu is zero on all measurable subsets of A. This is denoted by \mu \perp \nu. A refined form of Lebesgue's decomposition theorem decomposes a singular measure into a singular continuous measure and a discrete measure. See below for examples. Examples on R''n'' As a particular case, a measure defined on the Euclidean space \R^n is called ''singular'', if it is singular with respect to the Lebesgue measure on this space. For example, the Dirac delta function is a singular measure. Example. A discrete measure. The Heaviside step function on the real line, H(x) \ \stackrel \begin 0, & x 0 but \delta_0(U) = 0. Example. A singular continuous measure. The Cantor distribution has a cumulative distribu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Measure
In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of two topological spaces, except that there can be many natural choices for the product measure. Let (X_1, \Sigma_1) and (X_2, \Sigma_2) be two measurable spaces, that is, \Sigma_1 and \Sigma_2 are sigma algebras on X_1 and X_2 respectively, and let \mu_1 and \mu_2 be measures on these spaces. Denote by \Sigma_1 \otimes \Sigma_2 the sigma algebra on the Cartesian product X_1 \times X_2 generated by subsets of the form B_1 \times B_2, where B_1 \in \Sigma_1 and B_2 \in \Sigma_2. This sigma algebra is called the ''tensor-product σ-algebra'' on the product space. A ''product measure'' \mu_1 \times \mu_2 (also denoted by \mu_1 \otimes \mu_2 by many authors) is defined to be a measure on the measurable space (X_1 \times X_2, \Sigma_1 \ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function ''space''. In linear algebra Let be a vector space over a field and let be any set. The functions → can be given the structure of a vector space over where the operations are defined pointwise, that is, for any , : → , any in , and any in , define \begin (f+g)(x) &= f(x)+g(x) \\ (c\cdot f)(x) &= c\cdot f(x) \end When the domain has additional structure, one might consider instead the subset (or subspace) of all such functions which respect that structure. For example, if is also a vector space over , the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Japan
Japan ( ja, 日本, or , and formally , ''Nihonkoku'') is an island country in East Asia. It is situated in the northwest Pacific Ocean, and is bordered on the west by the Sea of Japan, while extending from the Sea of Okhotsk in the north toward the East China Sea, Philippine Sea, and Taiwan in the south. Japan is a part of the Ring of Fire, and spans Japanese archipelago, an archipelago of List of islands of Japan, 6852 islands covering ; the five main islands are Hokkaido, Honshu (the "mainland"), Shikoku, Kyushu, and Okinawa Island, Okinawa. Tokyo is the Capital of Japan, nation's capital and largest city, followed by Yokohama, Osaka, Nagoya, Sapporo, Fukuoka, Kobe, and Kyoto. Japan is the List of countries and dependencies by population, eleventh most populous country in the world, as well as one of the List of countries and dependencies by population density, most densely populated and Urbanization by country, urbanized. About three-fourths of Geography of Japan, the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shizuo Kakutani
was a Japanese-American mathematician, best known for his eponymous fixed-point theorem. Biography Kakutani attended Tohoku University in Sendai, where his advisor was Tatsujirō Shimizu. At one point he spent two years at the Institute for Advanced Study in Princeton at the invitation of the mathematician Hermann Weyl. While there, he also met John von Neumann. Kakutani received his Ph.D. in 1941 from Osaka University and taught there through World War II. He returned to the Institute for Advanced Study in 1948, and was given a professorship by Yale in 1949, where he won a students' choice award for excellence in teaching. Kakutani received two awards of the Japan Academy, the Imperial Prize and the Academy Prize in 1982, for his scholarly achievements in general and his work on functional analysis in particular. He was a Plenary Speaker of the ICM in 1950 in Cambridge, Massachusetts. Kakutani was married to Keiko ("Kay") Uchida, who was a sister to author Yoshiko Uc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the Germany, German mathematician Carl Friedrich Gauss. One reason why Gaussian measures are so ubiquitous in probability theory is the central limit theorem. Loosely speaking, it states that if a random variable ''X'' is obtained by summing a large number ''N'' of independent random variables of order 1, then ''X'' is of order \sqrt and its law is approximately Gaussian. Definitions Let ''n'' ∈ N and let ''B''0(R''n'') denote the complete measure, completion of the Borel sigma algebra, Borel ''σ''-algebra on R''n''. Let ''λ''''n'' : ''B''0(R''n'') → [0, +∞] denote the usual ''n''-dimensional Lebesgue measure. Then the standard Gaussian measure ''γ''''n'' : ''B''0(R''n'') → [0, 1] is defined by :\gamma^ (A) = \frac \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 changes under translation by certain elements of the Cameron–Martin Hilbert space. Motivation The standard Gaussian measure \gamma^n on n-dimensional Euclidean space \mathbf^n is not translation-invariant. (In fact, there is a unique translation invariant Radon measure up to scale by Haar's theorem: the n-dimensional Lebesgue measure, denoted here dx.) Instead, a measurable subset A has Gaussian measure :\gamma_n(A) = \frac\int_A \exp\left(-\tfrac12\langle x, x\rangle_\right)\,dx. Here \langle x,x\rangle_ refers to the standard Euclidean dot product in \mathbf^n. The Gaussian measure of the translation of A by a vector h \in \mathbf^n is :\begin \gamma_n(A-h) &= \frac\int_A \exp\left(-\tfrac12\langle x-h, x-h\rangle_\right)\,dx\\ pt&=\frac\int_A \exp\left(\frac\right)\exp\l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Measure Theory")
