a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are...

{\displaystyle X} forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on X {\displaystyle X} is the smallest σ-algebra containing all...

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...

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...

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...

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

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...

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

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...

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...

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

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...

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

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

quaternions i, j, and k, discovered in 1843: i = − σ 2 σ 3 = − i σ 1 j = − σ 3 σ 1 = − i σ 2 k = − σ 1 σ 2 = − i σ 3 {\displaystyle {\begin{aligned}\mathbf {i}...

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

{\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...

probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies measure properties such as countable additivity. The difference...

theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is finite on all...

Borel σ-algebra; for example, the norm topology and the weak topology on a separable Hilbert space lead to the same Borel σ-algebra. Not every σ-algebra is...

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...

σ-algebra B ( X ) {\displaystyle {\mathcal {B}}(X)} : is generated by the open sets of T {\displaystyle {\mathcal {T}}} . the cylindrical σ-algebra E...

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...

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

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

any set Ω {\displaystyle \Omega \,} (also called sample space) and a σ-algebra F {\displaystyle {\mathcal {F}}\,} on it, a measure P {\displaystyle P\...

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...

Lebesgue-measurable. The set of all such E {\displaystyle E} forms a σ-algebra. The Lebesgue measure of such a set is defined as its Lebesgue outer measure:...