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...

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...

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...

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...

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...

algebra of sets known as σ-algebra (aka σ-field). Sigma algebra also includes terms such as: σ(A), denoting the generated sigma-algebra of a set A Σ-finite...

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...

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...

forms a σ-algebra, i.e., it is closed under the operations of countable unions, countable intersections, and complementation. This σ-algebra is the...

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...

In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle...

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...

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...

on the σ-algebra σ ( F ) {\displaystyle \sigma ({\mathcal {F}})} generated by F . {\displaystyle {\mathcal {F}}.} If μ {\displaystyle \mu } is σ-finite...

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...

additional structure, by defining an algebra homomorphism Δ : A → A ⊗ A, and defining the tensor product representation as ρ = ( σ ⊗ τ ) ∘ Δ . {\displaystyle \rho...

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...

In database theory, relational algebra is a theory that uses algebraic structures for modeling data, and defining queries on it with well founded semantics...

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...

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...

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...

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...

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...

{\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...

respect to the σ-algebra generated by all left-continuous adapted processes. This σ-algebra is also called the predictable σ-algebra. Every deterministic...

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), ∩, ∪, ′...

𝜎-algebras. The theorem says that the smallest monotone class containing an algebra of sets G {\displaystyle G} is precisely the smallest 𝜎-algebra...

{\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...

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...

mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A {\displaystyle A} over the real or complex...