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...

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...

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...

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...

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...

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

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...

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

Sigma (/ˈsɪɡmə/ SIG-mə; uppercase Σ, lowercase σ, lowercase in word-final position ς; Greek: σίγμα) is the eighteenth letter of the Greek alphabet. In...

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...

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...

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...

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...

the eventual result). σ A ( R ) = σ A σ A ( R ) {\displaystyle \sigma _{A}(R)=\sigma _{A}\sigma _{A}(R)\,\!} σ A σ B ( R ) = σ B σ A ( R ) {\displaystyle...

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

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...

to use a σ-algebra, that is, a family closed under complementation and countable unions of its members. The most natural choice of σ-algebra is the Borel...

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...

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...

X\rangle ]} . Then, for any measurable convex function φ and any sub-σ-algebra G {\displaystyle {\mathfrak {G}}} of F {\displaystyle {\mathfrak {F}}}...

(X,{\mathcal {F}})} is called a σ-field of sets and the algebra F {\displaystyle {\mathcal {F}}} is called a σ-algebra if the following additional condition...

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...

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}...

for the σ-algebra E {\displaystyle {\mathcal {E}}} is the Borel σ-algebra B ( E ) {\displaystyle {\mathcal {B}}(E)} , which is the σ-algebra generated...

{\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...

(E)=\lambda ^{\!*\!}(E)} . The set of all such E {\displaystyle E} forms a σ-algebra. A set E {\displaystyle E} that does not satisfy the Carathéodory criterion...

q ) = { σ : { 1 , … , p + q } → { 1 , … , p + q } ∣ σ  is bijective , σ ( 1 ) < σ ( 2 ) < ⋯ < σ ( p ) , and  σ ( p + 1 ) < σ ( p + 2 ) < ⋯ < σ ( m ) }...

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