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Longer titles found: Cylindrical σ-algebra (view), Σ-Algebra of τ-past (view)

searching for Σ-algebra 66 found (172 total)

alternate case: σ-algebra

Kolmogorov's zero–one law (896 words) [view diff] exact match in snippet view article find links to article

_{k=n}^{\infty }F_{k}{\bigg )}} be the smallest σ-algebra containing Fn, Fn+1, .... The terminal σ-algebra of the Fn is defined as T ( ( F n ) n ∈ N ) =

{\displaystyle i\in I} let F i {\displaystyle {\mathcal {F}}_{i}} be a sub-σ-algebra of A {\displaystyle {\mathcal {A}}} . Then F := ( F i ) i ∈ I {\displaystyle

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

the open sets. If we take the natural filtration F•X, where FtX is the σ-algebra generated by the pre-images Xs−1(B) for Borel subsets B of R and times

X_{j}^{-1}(A)\right|j\in I,j\leq i,A\in \Sigma \right\},} i.e., the smallest σ-algebra on Ω that contains all pre-images of Σ-measurable subsets of S for "times"

cylinder sets. These cylinder sets generate a σ-algebra, the Borel σ-algebra; this is the smallest σ-algebra that contains the topology. Define a function

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

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

Lebesgue measurable sets, are a σ-algebra F ; {\displaystyle \textstyle {\mathcal {F}};} it contains the Lebesgue σ-algebra and Z . {\displaystyle \textstyle

law for exchangeable sequences, Kolmogorov's zero–one law for the tail σ-algebra, Lévy's zero–one law, related to martingale convergence. Topological zero–one

and B ( R ) {\displaystyle {\mathcal {B}}(\mathbb {R} )} is its Borel σ-algebra, then the definition of random element is the classical definition of

{\displaystyle Y} are independent, since the σ-algebra generated by a constant random variable is the trivial σ-algebra { ∅ , Ω } {\displaystyle \{\varnothing

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

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

a σ-algebra of subsets of X {\displaystyle X} and P {\displaystyle \mathbb {P} } is a probability measure defined on that σ-algebra. Sets in the σ-algebra

{\displaystyle E(f|{\mathcal {C}})} is the conditional expectation given the σ-algebra C {\displaystyle {\mathcal {C}}} of invariant sets of T. Corollary (Pointwise

invariant Borel probability measure. That is, if Borel(X) denotes the Borel σ-algebra generated by the collection T of open subsets of X, then there exists

{\displaystyle R} and B {\displaystyle B} are conditionally independent given a σ-algebra Σ {\displaystyle \Sigma } if Pr ( R , B ∣ Σ ) = Pr ( R ∣ Σ ) Pr ( B ∣

moment. Let (M, d) be a metric space, and let B(M) be the Borel σ-algebra on M, the σ-algebra generated by the d-open subsets of M. (For technical reasons

of all open sets in R n {\displaystyle \mathbb {R} ^{n}} ) the Borel σ-algebra on R {\displaystyle \mathbb {R} } (i.e. the set of all Borel sets in R

particular, the set of events on which probability is defined may be some σ-algebra on S {\displaystyle S} and not necessarily the full power set. Atom (measure

{\displaystyle (X,M)} a measurable space with M {\displaystyle M} a Borel σ-algebra on X {\displaystyle X} . A POVM is a function F {\displaystyle F} defined

particular, the set of events on which probability is defined may be some σ-algebra on S {\displaystyle S} and not necessarily the full power set. In some

together with the Hamiltonian, are used to define a measure on the Borel σ-algebra in the following way: The measure of a cylinder set, i.e. an element of

Shannon's basic measures of information are necessary to deal with the σ-algebra generated by the sets that would be associated to three or more arbitrary

those observations" it is meant that Ŷt is measurable with respect to the σ-algebra Gt generated by the observations Zs, 0 ≤ s ≤ t. Denote by K = K(Z, t)

{\displaystyle (B,{\mathcal {B}})} is a measurable space whose underlying σ-algebra is discrete, so in particular contains singleton sets of B {\displaystyle

Borel sets of a topological space X consists of all sets in the smallest σ-algebra containing the open sets of X. This means that the Borel sets of X are

filtration of Brownian motion B : [0, T] × Ω → R: Σt is the smallest σ-algebra containing all Bs−1(A) for times 0 ≤ s ≤ t and Borel sets A ⊆ R; E[·|Σt]

proper. (See the article on topological groups.) In a Polish group G, the σ-algebra of Haar null sets satisfies the countable chain condition if and only

be a Polish space, B ( X ) {\displaystyle {\mathcal {B}}(X)} the Borel σ-algebra of X {\displaystyle X} , ( Ω , F ) {\displaystyle (\Omega ,{\mathcal {F}})}

X {\displaystyle X} , and if B {\displaystyle {\mathcal {B}}} is the σ-algebra of Borel sets, if T {\displaystyle T} is μ {\displaystyle \mu } -ergodic

if X is a Polish space and B {\displaystyle {\mathcal {B}}} its Borel σ-algebra, C l ( X ) {\displaystyle \mathrm {Cl} (X)} is the set of nonempty closed

j=1}^{n}|\sigma _{ij}|^{2}.} Let Z be a random variable that is independent of the σ-algebra generated by Bs, s ≥ 0, and with finite second moment: E [ | Z | 2 ] <

{\displaystyle X} be a real vector space, A {\displaystyle {\mathcal {A}}} be σ-algebra that is invariant under translation by vectors h ∈ X {\displaystyle h\in

collection of subsets of X under countably many set operations is called the σ-algebra generated by the collection. In the preceding sections, closures are considered

R + {\displaystyle \mathbb {R} ^{+}} is the restriction of the Borel σ-algebra B ( R ) {\displaystyle {\mathfrak {B}}(\mathbb {R} )} to R + {\displaystyle

probability space by letting F t {\displaystyle {\mathcal {F}}_{t}} be the σ-algebra generated by all the sets of the form ( B s ) − 1 ( A ) {\displaystyle

Boolean algebra. When the measure space is the unit interval with the σ-algebra of Lebesgue measurable sets, the Boolean algebra is called the random

of their symmetric difference. If μ is a σ-finite measure defined on a σ-algebra Σ, the function d μ ( X , Y ) = μ ( X Δ Y ) {\displaystyle d_{\mu }(X

\mathbb {L} } , F Λ {\displaystyle {\mathcal {F}}_{\Lambda }} is the σ-algebra generated by the family of functions ( σ ( t ) ) t ∈ Λ {\displaystyle

the Banach space of bounded operators on H. A mapping E from the Borel σ-algebra on X to L ( H ) {\displaystyle L(H)} is called an operator-valued measure

N } Z {\displaystyle \mu =\{p_{1},\ldots ,p_{N}\}^{\mathbb {Z} }} The σ-algebra A {\displaystyle {\mathcal {A}}} on X is the product sigma algebra; that

is, the Borel σ-algebra on the real numbers (which is generated by all real intervals) is distinct from the Lebesgue-measure σ-algebra on the real numbers

{A}}.} Suppose that A {\displaystyle \scriptstyle {\mathcal {A}}} is a σ-algebra. If for every sequence A 1 , A 2 , … , A n , … {\displaystyle A_{1},A_{2}

2}}\int _{E}{\frac {dx}{1+x}}.} Then μ is a probability measure on the σ-algebra of Borel subsets of I. The measure μ is equivalent to the Lebesgue measure

probability distributions where A {\displaystyle A} is a set provided with some σ-algebra of measurable subsets. In particular we can take A {\displaystyle A} to

{\displaystyle F} , and define F ∞ {\displaystyle F_{\infty }} to be the minimal σ-algebra generated by ( F k ) k ∈ N {\displaystyle (F_{k})_{k\in \mathbf {N} }}

{\displaystyle r} (usually r {\displaystyle r} is a Lebesgue measure on a Borel σ-algebra). Let P {\displaystyle P} and Q {\displaystyle Q} be probability density

endowed with a topological structure. Consider the Borel measure (on the σ-algebra generated by the open sets) in the state space of each random variable

separable Banach space with the norm induced topology, we use the Borel σ-algebra and denote the dual space as E ∗ {\displaystyle E^{*}} . Let ⟨ z , S ⟩

complement and countable union. That is, the Borel sets are the smallest σ-algebra of subsets of Aω containing all the open sets. The Borel sets are classified

for a topology on X if and only if it covers X. By definition, every σ-algebra, every filter (and so in particular, every neighborhood filter), and every

for all t and h ≥ 0, the conditional expectation conditioned on the σ-algebra Σt and the expectation of the process "restarted" from Xt satisfy the

countable Hausdorff space and S {\displaystyle {\mathcal {S}}} is its Borel σ-algebra. Consider now an integer-valued locally finite kernel ξ {\displaystyle

considered pathological, L∞(X) is not a von Neumann algebra; for example, the σ-algebra of measurable sets might be the countable-cocountable algebra on an uncountable

process, B ( [ 0 , T ] ) {\displaystyle {\mathcal {B}}([0,T])} the Borel σ-algebra, ∫ f d W t {\displaystyle \int f\;dW_{t}} be the Wiener integral, d t

{\displaystyle {\mathcal {F}}_{0+}^{X}} is the P {\displaystyle P} -trivial σ-algebra, i.e. all events have probability 0 {\displaystyle 0} or 1 {\displaystyle

the space of measures of bounded variation. The space of measures on a σ-algebra of sets is a Banach space, called the ca space, relative to this norm

defined on I and let μ be a continuous non-negative measure on the Borel σ-algebra of I satisfying μ([a, t]) < ∞ for all t ∈ I (this is certainly satisfied

"Sufficiently regular subset" here means a Borel set; that is, an element of the σ-algebra generated by the compact sets. More precisely, a right Haar measure on

of "open" sets; or the structure of a measurable space, treated as the σ-algebra of "measurable" sets; both are elements of P(P(X)). These are second-order

{\displaystyle {\mathcal {M}}} is an uncertain measure on the product σ-algebra satisfying M { ∏ i = 1 n Λ i } = min 1 ≤ i ≤ n M i { Λ i } {\displaystyle

spaces associated to measure spaces (X, M, μ), where X is a set, M is a σ-algebra of subsets of X, and μ is a countably additive measure on M. Let L2(X

{\displaystyle (X,M)} is a measurable space with M {\displaystyle M} a Borel σ-algebra on X {\displaystyle X} and let F {\displaystyle F} be a POVM from M {\displaystyle

sets. Also called a set of first category. measure 1.  A measure on a σ-algebra of subsets of a set 2.  A probability measure on the algebra of all subsets