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...
30 KB (5,339 words) - 13:11, 26 March 2024
Borel set (redirect from Borel σ-algebra)
Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on X is the smallest σ-algebra containing all open sets...
13 KB (1,793 words) - 02:48, 4 June 2024
Kolmogorov's zero–one law (redirect from Tail σ-algebra)
specifies that a certain type of event, namely a tail event of independent σ-algebras, will either almost surely happen or almost surely not happen; that is...
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In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle...
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theory and functional analysis — the cylindrical σ-algebra or product σ-algebra is a type of σ-algebra which is often used when studying product measures...
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space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. It captures and generalises...
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conditional expectation was generalized to its modern definition using sub-σ-algebras. If A is an event in F {\displaystyle {\mathcal {F}}} with nonzero probability...
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Cocountability (redirect from Countable-cocountable algebra)
forms a σ-algebra, i.e., it is closed under the operations of countable unions, countable intersections, and complementation. This σ-algebra is the...
2 KB (225 words) - 01:06, 8 April 2024
simpler to work with than 𝜎-algebras. For example, it may be awkward to work with 𝜎-algebras generated by infinitely many sets σ ( E 1 , E 2 , … ) . {\displaystyle...
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analysis we want to do. This leads to different choices of sample space. The σ-algebra F {\displaystyle {\mathcal {F}}} is a collection of all the events we...
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sense) if and only if the σ-algebras that they generate are independent (in the new sense). The σ-algebra generated by an event E ∈ Σ {\displaystyle E\in \Sigma...
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The σ-algebra of τ-past, (also named stopped σ-algebra, stopped σ-field, or σ-field of τ-past) is a σ-algebra associated with a stopping time in the theory...
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on the σ-algebra σ ( F ) {\displaystyle \sigma ({\mathcal {F}})} generated by F . {\displaystyle {\mathcal {F}}.} If μ {\displaystyle \mu } is σ-finite...
42 KB (7,477 words) - 01:50, 6 May 2024
Sigma-ideal (redirect from Σ-ideal)
In mathematics, particularly measure theory, a 𝜎-ideal, or sigma ideal, of a σ-algebra (𝜎, read "sigma") is a subset with certain desirable closure properties...
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additional structure, by defining an algebra homomorphism Δ : A → A ⊗ A, and defining the tensor product representation as ρ = ( σ ⊗ τ ) ∘ Δ . {\displaystyle \rho...
30 KB (4,256 words) - 10:18, 18 August 2024
Pauli matrices (redirect from Pauli algebra)
anticommutator gives [ σ j , σ k ] + { σ j , σ k } = ( σ j σ k − σ k σ j ) + ( σ j σ k + σ k σ j ) 2 i ε j k ℓ σ ℓ + 2 δ j k I = 2 σ j σ k {\displaystyle...
44 KB (7,416 words) - 16:25, 12 July 2024
In database theory, relational algebra is a theory that uses algebraic structures for modeling data, and defining queries on it with well founded semantics...
51 KB (6,333 words) - 19:48, 15 August 2024
Real-valued function (section Algebraic structure)
a partially ordered ring. The σ-algebra of Borel sets is an important structure on real numbers. If X has its σ-algebra and a function f is such that...
8 KB (993 words) - 15:40, 22 June 2023
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 ∞). A set A in Σ is of finite measure...
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{\mathfrak {B}}(X)} be the smallest σ-algebra that contains the open sets of X {\displaystyle X} ; this is known as the σ-algebra of Borel sets. A Borel measure...
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Predictable process (redirect from Predictable σ-algebra)
respect to the σ-algebra generated by all left-continuous adapted processes. This σ-algebra is also called the predictable σ-algebra. Every deterministic...
3 KB (273 words) - 21:37, 5 July 2023
Linear algebra is the branch of mathematics concerning linear equations such as: a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b...
64 KB (7,780 words) - 04:31, 11 August 2024
we would take the 𝜎-algebra on R 2 {\displaystyle \mathbb {R} ^{2}} to be B ⊗ B , {\displaystyle B\otimes B,} the smallest 𝜎-algebra containing all measurable...
6 KB (826 words) - 17:41, 19 March 2024
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the...
23 KB (4,160 words) - 15:09, 12 January 2024
topomorphisms between σ-complete interior algebras.) Given a topological space X = ⟨X, T⟩ one can form the power set Boolean algebra of X: ⟨P(X), ∩, ∪, ′...
30 KB (3,849 words) - 16:32, 8 April 2024
𝜎-algebras. The theorem says that the smallest monotone class containing an algebra of sets G {\displaystyle G} is precisely the smallest 𝜎-algebra...
5 KB (694 words) - 12:02, 7 September 2023
{\displaystyle A^{c}=X\setminus A.} A σ-algebra is an algebra that is also closed under countable unions, or equivalently a σ-ring that contains X . {\displaystyle...
8 KB (1,426 words) - 05:09, 22 April 2024
In mathematics, the algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets...
15 KB (1,865 words) - 16:28, 28 May 2024
mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A {\displaystyle A} over the real or complex...
17 KB (2,602 words) - 13:01, 17 July 2024