probability theory, a σ-algebra ("sigma algebra"; also σ-field, where the σ comes from the German "Summe") on a set X is a nonempty collection Σ of subsets of...

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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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...

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

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

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...

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...

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...

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...

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...

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

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...

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), ∩, ∪, ′...

{\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...

{\displaystyle \Omega } satisfying a set of axioms weaker than those of 𝜎-algebra. Dynkin systems are sometimes referred to as 𝜆-systems (Dynkin himself...

Hahn–Kolmogorov theorem. Let Σ 0 {\displaystyle \Sigma _{0}} be an algebra of subsets of a set X . {\displaystyle X.} Consider a set function μ 0 : Σ 0 → [ 0 , ∞ ] {\displaystyle...

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...

{\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...

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