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Cylinder set

In mathematics, a cylinder set is a subset of a product space X = \prod_{\alpha \in A} X_\alpha, where A is an index set, defined as the set of all points that lie in a specified subset U of the finite-dimensional product \prod_{\alpha \in S} X_\alpha for some finite subset S \subset A, extended by the full spaces X_\alpha for all \alpha \notin S; formally, it takes the form U \times \prod_{\alpha \notin S} X_\alpha.^[1] These sets are essential in both topology and measure theory, serving as a basis for the product topology on infinite products, where open sets are unions of cylinder sets with open U, ensuring that projection maps to finite coordinates are continuous.^[2] In measure theory, cylinder sets generate the product \sigma-algebra \bigotimes_{\alpha \in A} \mathcal{F}_\alpha, the smallest \sigma-algebra making all coordinate projections measurable, which is crucial for constructing product measures on infinite-dimensional spaces via Kolmogorov's extension theorem.^[1] For example, in \mathbb{R}^\mathbb{N}, a one-dimensional cylinder set might be \mathbb{R} \times [-1, 2) \times \mathbb{R} \times \cdots, restricting the second coordinate while allowing others to vary freely.^[1] Cylinder sets also play a key role in probability theory for defining measures on spaces of sequences or functions, such as Gaussian measures on Hilbert spaces, by specifying finite-dimensional marginals.^[1]

Introduction and History

Historical Development

The name "cylinder set" derives from the geometric analogy to cylinders in finite-dimensional spaces, which are unbounded in certain directions while constrained in others. This intuition was adapted in early 20th-century measure theory as mathematicians sought to extend measures from finite to infinite products. The modern mathematical formalization of cylinder sets occurred in Andrey Kolmogorov's 1933 book Foundations of the Theory of Probability, where they were introduced to construct the product σ-algebra on infinite-dimensional spaces, ensuring consistency for probability measures defined via finite-dimensional distributions. Kolmogorov defined cylinder sets as subsets of the space R^M (with M an infinite index set) depending on finitely many coordinates, forming a field that generates the Borel σ-algebra for infinite sequences of random variables, thus enabling the rigorous treatment of stochastic processes. In the post-1930s era, the concept extended to topological vector spaces through Laurent Schwartz's work. His theory of distributions from 1945–1950 provided a framework for generalized functions in infinite dimensions, and later work, including his 1973 book Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures, incorporated cylinder sets to underpin cylindrical measures—finitely additive set functions on algebras generated by finite-dimensional projections—crucial for defining Radon measures in infinite dimensions.^[3] By the 1950s, cylinder sets achieved broader formalization in abstract measure theory, as seen in Paul Halmos's Measure Theory (1950), which detailed their role in product measures and σ-algebras for infinite spaces, emphasizing countable additivity extensions. Similarly, Lynn H. Loomis's An Introduction to Abstract Harmonic Analysis (1953) incorporated cylinder sets in the analysis of infinite product groups and measures, solidifying their place in harmonic and topological contexts.^[4] These milestones marked the transition from probabilistic origins to a versatile tool in functional analysis.

Motivations and Basic Intuition

Cylinder sets provide a fundamental tool for analyzing infinite-dimensional spaces, such as infinite products of measurable sets, by focusing on dependencies among only finitely many coordinates at a time. This approach circumvents the challenges posed by uncountable products, where direct definitions of topologies or measures become intractable due to the vast cardinality involved. By "slicing" the space into subsets determined by finite projections, cylinder sets enable the extension of finite-dimensional structures—like measures or open sets—to the full infinite-dimensional setting, ensuring consistency across dimensions. Geometrically, the intuition for cylinder sets draws from familiar shapes in lower dimensions. In \mathbb{R}^2, considered as \mathbb{R} \times \mathbb{R}, a basic cylinder set resembles a vertical strip, such as an open interval (a, b) on the x-axis extended across the entire y-axis, forming (a, b) \times \mathbb{R}. This "cylinder" is unbounded in one direction while constrained in the other, generalizing in higher dimensions to sets that are restricted in finitely many coordinates and unrestricted (i.e., the full space) in all others. Such constructions form the basis for the product topology, allowing intuitive visualization of open sets as finite "tubes" embedded in an otherwise free infinite expanse.^[5] A concrete example illustrates their utility in probability: consider the space [0,1]^\infty, modeling an infinite sequence of independent uniform random variables, akin to infinite coin flips with fair probability $1/2 for heads (1) or tails (0). A cylinder set here specifies outcomes for the first k flips—say, the set of sequences starting with 1, 0, 1—while leaving the remaining infinite flips arbitrary; its probability is simply (1/2)^k = 1/8 for k=3, reflecting the independence of the coordinates. This finite specification captures events of practical interest without needing to address the entire infinite tail, making cylinder sets indispensable for defining product measures on sequence spaces.^[6] These concepts build essential intuition for subsequent developments, highlighting why finite intersections of projections naturally generate the relevant structures for topologies and measures in product spaces. As pioneered by Kolmogorov in his extension theorem, cylinder sets ensure that measures defined on finite-dimensional marginals can be consistently lifted to the infinite case, providing a prerequisite foundation for rigorous constructions.^[7]

Definitions

General Definition in Product Spaces

In a product space X = \prod_{i \in I} X_i, where I is an arbitrary index set (possibly infinite) and each X_i is a nonempty set, a cylinder set is a subset of the form

C = \bigcap_{j=1}^n \pi_{i_j}^{-1}(A_j),

where n \in \mathbb{N} is finite, i_1, \dots, i_n are distinct elements of I, each \pi_{i_j}: X \to X_{i_j} is the canonical projection map, and each A_j \subseteq X_{i_j}. When each factor space X_i is endowed with a topology \mathcal{T}_i, the collection of all cylinder sets for which each A_j is open in X_{i_j} (with respect to \mathcal{T}_{i_j}) forms a basis for the product topology on X, also known as the Tychonoff topology. This basis ensures that all projection maps \pi_i are continuous and that the product topology is the coarsest topology making this property hold.^[8] If instead each X_i carries a \sigma-algebra \mathcal{M}_i, then the collection of cylinder sets where each A_j \in \mathcal{M}_{i_j} generates the product \sigma-algebra on X, denoted \bigotimes_{i \in I} \mathcal{M}_i and also referred to as the cylinder \sigma-algebra. This \sigma-algebra is the smallest one containing all sets of the form \pi_i^{-1}(B) for B \in \mathcal{M}_i and i \in I, with finite intersections thereof also belonging to the generating family. The product topology differs from the box topology on X, whose basis consists of arbitrary products \prod_{i \in I} U_i with each U_i open in X_i; these basis elements correspond to (possibly infinite) intersections of cylinder sets, whereas standard cylinder sets require only finite intersections.^[8] For infinite index sets I, the box topology is strictly finer than the product topology, as the latter does not include all such infinite products as open sets.^[8]

Cylinder Sets in Products of Discrete Sets

In the context of symbolic dynamics, cylinder sets arise naturally in the product space X = S^{\mathbb{Z}}, where S is a finite discrete set serving as the alphabet, and elements of X are bi-infinite sequences x = (x_i)_{i \in \mathbb{Z}} with each x_i \in S. This space is equipped with the product topology, making it a compact metric space suitable for modeling discrete dynamical systems. The basic cylinder sets in this setting are defined as C_t(a) = \{ x \in X : x_t = a \} for some integer time index t \in \mathbb{Z} and symbol a \in S; these sets consist of all sequences that fix the coordinate at position t to the value a.^[9] More generally, finite cylinder sets are finite intersections of basic cylinders, denoted C_{t_1, \dots, t_n}(a_1, \dots, a_n) = \{ x \in X : x_{t_j} = a_j \ \forall j = 1, \dots, n \}, where t_1 < \cdots < t_n are distinct integers and a_j \in S. These specify finite patterns or blocks of symbols at specified positions in the bi-infinite sequences, capturing local constraints on the global structure of elements in X.^[10] In the product topology, these finite cylinders are clopen sets—both open and closed—because they are finite intersections of basic open sets and their complements are unions of similar cylinders.^[11] Moreover, the collection of all finite cylinder sets forms a basis for the topology on X, meaning every open set can be expressed as a union of such cylinders, and they generate the Borel σ-algebra on X, which is crucial for measure-theoretic constructions in this discrete framework.^[9] A key application of cylinder sets occurs in the study of shift spaces, particularly subshifts of finite type (SFTs), which are closed shift-invariant subsets of X defined by prohibiting certain finite patterns. In an SFT, forbidden blocks correspond to empty cylinder sets; for instance, if the transition matrix A for symbols in S = \{1, \dots, k\} has zeros indicating disallowed consecutive symbols, then cylinders like C_{t,t+1}(i,j) are empty whenever A_{i,j} = 0, ensuring no sequence contains the forbidden block ij.^[11] This structure allows SFTs to be represented as path spaces on directed graphs, where edges correspond to allowed symbols and cylinders delineate admissible itineraries. For example, the even shift SFT over S = \{0,1\}, defined by forbidding the block "11", excludes all cylinders C_{t,t+1}(1,1), resulting in sequences with no two consecutive 1s.^[10]

Definition for Vector Spaces

In the context of vector spaces, cylinder sets are defined using the dual space to capture finite-dimensional projections of infinite-dimensional structures. For a vector space V over a field K (typically \mathbb{R} or \mathbb{C}), the algebraic dual V^* consists of all linear functionals f: V \to K. A cylinder set is then specified by a finite collection of such functionals f_1, \dots, f_n \in V^* and a Borel subset A \subseteq K^n, given by

C_A(f_1, \dots, f_n) = \{ x \in V : (f_1(x), \dots, f_n(x)) \in A \}.

This construction generalizes the notion of cylinders from product spaces to linear settings, where the map x \mapsto (f_1(x), \dots, f_n(x)) is a linear surjection onto K^n.^[12] In topological vector spaces, particularly locally convex ones, the definition is refined to ensure compatibility with the topology by restricting to the continuous dual V' \subseteq V^*, comprising continuous linear functionals. Thus, topological cylinder sets are of the form C_A(\ell_1, \dots, \ell_n) = \{ x \in V : (\ell_1(x), \dots, \ell_n(x)) \in A \} with \ell_j \in V' and A Borel in K^n. These sets arise as preimages under finite-dimensional continuous linear maps L: V \to K^n, L(x) = (\ell_1(x), \dots, \ell_n(x)), and form a basis for the weak topology \sigma(V, V'), where neighborhoods are defined by finite constraints on continuous functionals. In spaces like the Schwartz space \mathcal{S}(\mathbb{R}^d) of rapidly decreasing functions, its dual \mathcal{S}'(\mathbb{R}^d) of tempered distributions uses cylinder sets based on pairings \langle u, \phi_j \rangle for test functions \phi_j \in \mathcal{S}(\mathbb{R}^d), enabling measure-theoretic constructions on infinite-dimensional spaces.^[13]^[14] Algebraic cylinder sets, using arbitrary elements of V^*, differ from topological ones by not necessarily respecting the topology, leading to coarser structures; the latter are crucial for defining weak and weak* topologies on dual pairs (V, V'). For instance, in the separable Hilbert space \ell^2 of square-summable sequences over \mathbb{R}, the standard orthonormal basis \{e_k\} induces continuous coordinate functionals \pi_k(x) = \langle x, e_k \rangle = x_k \in V', so cylinder sets include C_A(\pi_1, \dots, \pi_n) = \{ x \in \ell^2 : (x_1, \dots, x_n) \in A \}, which generate the product topology restricted to \ell^2. This framework underpins analyses in functional analysis, such as Gaussian measures on Banach spaces.^[12]^[15]

Properties

Topological Properties

Cylinder sets play a fundamental role in defining the product topology on the Cartesian product of topological spaces. Specifically, for a product space \prod_{i \in I} X_i, the collection of all sets of the form \pi_i^{-1}(U_i) = U_i \times \prod_{j \neq i} X_j, where U_i is open in X_i and i \in I, forms a subbasis for the product topology.^[16]^[17] The finite intersections of these subbasic sets, known as cylinder sets with finitely many specified coordinates, constitute a basis for the topology, ensuring that every open set in the product space can be expressed as an arbitrary union of such cylinder sets.^[18]^[16] This structure guarantees that the product topology is the coarsest topology making all projection maps \pi_i: \prod_{i \in I} X_i \to X_i continuous.^[17] In general, cylinder sets are open in the product topology whenever the sets specifying the coordinates are open in their respective factor spaces.^[18] For products of discrete spaces, cylinder sets are clopen, as the discrete topology renders singletons both open and closed, and their preimages under projections inherit this property.^[19] More broadly, the product topology induced by cylinder sets is the initial topology with respect to the family of projection maps, meaning it is the coarsest topology such that all projections remain continuous.^[16] This contrasts with the box topology, which is generated by all possible products of open sets from the factors (allowing infinitely many non-trivial specifications) and is strictly finer than the product topology for infinite index sets.^[16] Cylinder sets are instrumental in establishing compactness properties of product spaces. In the context of Tychonoff's theorem, the compactness of the product \prod_{i \in I} X_i when each X_i is compact follows from the fact that the basic open cylinder sets satisfy the finite intersection property, allowing the transfer of compactness from finite subproducts to the entire space via the tube lemma or Alexander's subbase theorem.^[17]^[18] This ensures that every open cover of the product has a finite subcover, leveraging the cylinder structure to cover the infinite-dimensional space effectively.^[16]

Measure-Theoretic Properties

In measure theory, cylinder sets play a fundamental role in constructing product measures on infinite product spaces. The collection of all cylinder sets, defined as sets of the form \prod_{i \in I} A_i where I is a finite subset of the index set and each A_i is measurable in the factor space, generates the product \sigma-algebra when the factors are equipped with \sigma-algebras.^[20] Specifically, the \sigma-algebra generated by these cylinder sets coincides with the product \sigma-algebra, which is the smallest \sigma-algebra containing all such cylinders.^[21] Moreover, the family of cylinder sets forms a \pi-system, as it is closed under finite intersections: the intersection of two cylinders depends only on the union of their finite coordinate sets, resulting in another cylinder.^[22] This \pi-system property, combined with the monotone class theorem or Dynkin's \pi-\lambda theorem, facilitates uniqueness results for measures defined on the product \sigma-algebra.^[23] Cylinder sets also possess a semi-ring structure, which is essential for measure construction via extension theorems. A semi-ring is a collection closed under finite intersections such that the difference of two sets can be expressed as a finite disjoint union of sets in the collection; for cylinder sets, this holds because the complement relative to a larger cylinder decomposes into disjoint sub-cylinders.^[24] This semi-ring property allows the application of Carathéodory's extension theorem to extend a finitely additive set function defined on cylinders—such as a pre-measure assigning volumes based on factor measures—to a unique \sigma-additive measure on the generated \sigma-algebra, provided the pre-measure is \sigma-finite and continuous from below.^[25] In product spaces, this extension ensures the construction of product measures like the infinite tensor product of probability measures on discrete factors.^[26] For the Kolmogorov extension theorem, consistency conditions on finite-dimensional distributions guarantee the existence of a process measure on the product space. These distributions are specified on cylinder sets corresponding to finite coordinate projections, and finite additivity on disjoint cylinders—arising from the disjointness in the finite product spaces—ensures that the set function on the algebra of finite cylinder unions is well-defined and extends to the full \sigma-algebra.^[7] The theorem requires that the finite-dimensional measures be consistent, meaning the marginals on subsets of coordinates match, which preserves finite additivity across overlapping cylinders without altering the measure values.^[27] Under group actions such as the shift map on product spaces like sequence spaces, cylinder sets exhibit invariance properties with respect to certain measures. The shift transformation maps a cylinder defined by coordinates in positions j+1 to j+n to another cylinder in positions j to j+n-1, preserving the class of cylinders.^[28] For invariant measures, such as stationary Markov or Bernoulli measures on symbolic dynamics, the measure of a cylinder is independent of its starting position, ensuring that the shift leaves cylinder measures unchanged.^[28] In broader ergodic settings, quasi-invariant measures maintain that a cylinder has positive measure if and only if its image under the shift does, allowing for Radon-Nikodym derivatives to relate the measures, though the absolute continuity may vary.^[29]

Applications

In Probability and Measure Theory

Cylinder sets play a fundamental role in probability and measure theory, particularly in the construction of measures on infinite product spaces. They form the basis for defining consistent families of finite-dimensional distributions that can be extended to full probability measures on the entire space. This is exemplified by the Kolmogorov extension theorem, which relies on the algebra generated by cylinder sets to ensure the existence of such measures. The Kolmogorov extension theorem states that if a sequence of probability spaces (\Omega_n, \mathcal{F}_n, P_n) for n = 1, 2, \dots satisfies consistency conditions—specifically, marginal distributions match appropriately—then there exists a probability measure P on the infinite product space \prod_{n=1}^\infty \Omega_n such that the finite-dimensional projections agree with the given P_n. More precisely, for spaces (E, \mathcal{E}), a consistent family of finite-dimensional distributions \mu_{t_1, \dots, t_n} on E^n for times $0 \leq t_1 < \dots < t_n < \infty extends to a probability measure on the space of functions from [0, \infty) to E equipped with the \sigma-algebra generated by cylinder sets, where a cylinder set is of the form \{ \omega \in \Omega : (\omega(t_1), \dots, \omega(t_n)) \in B \} for B \in \mathcal{E}^{\otimes n}. The proof outline proceeds by first defining a finitely additive set function on the algebra of cylinder sets via the finite-dimensional distributions, which satisfies the consistency conditions. Carathéodory's extension theorem then applies, provided the set function is countably additive and continuous at the empty set; this is verified by showing that the measure of decreasing sequences of cylinder sets converging to the empty set approaches zero, leveraging the tightness or regularity of the finite-dimensional measures. The resulting measure on the \sigma-algebra generated by cylinders defines the full probability space. Cylinder set measures serve as prototypes for infinite-dimensional measures, where the measure is specified on cylinder sets depending on finitely many coordinates and extended uniquely. A prominent example is Gaussian cylinder measures on Banach spaces, defined by specifying centered Gaussian distributions to finite-dimensional projections onto subspaces spanned by finitely many basis elements, with the covariance given by the restriction of the covariance operator K, ensuring consistency across dimensions. For instance, on a separable Banach space B, for a cylinder C = \pi^{-1}(B) where \pi: B \to V is the projection onto a finite-dimensional subspace V and B \subset V is Borel, \mu(C) is the Gaussian measure of B on V with covariance K|_V; the measure extends to the full \sigma-algebra if the covariance K is nuclear.^[30] Specific examples illustrate these constructions. The Bernoulli measure on the space \{0,1\}^\mathbb{N}, equipped with the product \sigma-algebra generated by cylinders, assigns probability (1/2)^n to any cylinder set specified by the first n coordinates, corresponding to the infinite product of fair coin toss measures; this extends via Kolmogorov to the full space and models sequences of independent Bernoulli trials. Similarly, p-adic measures on the p-adic integers \mathbb{Z}_p use cylinder sets defined by congruence classes modulo p^n, with the Haar measure assigning p^{-n} to such cylinders of level n, providing a canonical probability measure on this totally disconnected space. In abstract Wiener spaces, cylinder sets define measures on a reproducing kernel Hilbert space H embedded continuously into a Banach space B, where the Gaussian measure on B is specified via finite-dimensional Gaussians on cylinders in H, ensuring the measure is supported on B despite H being dense but meager. This framework, introduced by Gross, allows rigorous treatment of Wiener measure on infinite-dimensional spaces without requiring separability in the Cameron-Martin sense directly on B.

In Dynamical Systems and Analysis

In symbolic dynamics, cylinder sets serve as the fundamental building blocks for the topology on shift spaces, particularly subshifts of finite type, where they form a basis of clopen sets in the product topology on the space of infinite sequences over a finite alphabet. These sets enable the construction of invariant measures and the computation of topological entropy, with the measure of maximal entropy—known as the Parry measure—for irreducible subshifts of finite type being uniquely determined by its values on the cylinder sets corresponding to the edges of the transition graph. For instance, in a full shift on two symbols, the Parry measure assigns equal probability to each basic cylinder of length n, yielding the Bernoulli measure with maximal entropy log 2. This framework underpins the study of mixing properties and symbolic representations of smooth dynamical systems, where cylinder sets approximate local behavior under the shift map. Cylinder sets also induce natural metrics on sequence spaces, turning the space of bi-infinite sequences into a compact metric space suitable for analyzing dynamical properties. A standard cylinder metric is defined by d(x, y) = 2^{-k}, where k is the smallest index such that the k-th coordinates of x and y differ, making each cylinder set a metric ball of radius 2^{-n} around its central sequence. This metric ensures that the shift map is continuous and expansive, facilitating proofs of uniqueness for measures of maximal entropy and the specification property in symbolic dynamics. Such metrics are essential for embedding theorems, like the one by Krieger, which classify subshifts up to conjugacy via their entropy and periodic points counted within cylinder neighborhoods. In analytical contexts, cylinder sets approximate functional integrals in Feynman path integrals for quantum mechanics, where the path space is equipped with a cylindrical σ-algebra generated by finite-dimensional projections, allowing the definition of Wiener measures on cylinder sets that extend to the full space under suitable regularity conditions. This cylindrical approximation enables rigorous treatment of the imaginary-time path integral, as in the case of the harmonic oscillator, where the measure on cylinders corresponds to Gaussian distributions on finite time slices. Similarly, in statistical mechanics, cylinder expansions compute partition functions for lattice models like the cyclic solid-on-solid (CSOS) model on cylindrical geometries, where fixed-boundary cylinder partition functions are evaluated via transfer matrix methods to determine critical exponents and phase transitions. Beyond these, cylinder sets feature in nonsingular odometers and Markov odometers, where partitions into cylinder sets generate the σ-algebra for non-singular transformations, enabling orbit equivalence classifications of ergodic systems via Bratteli-Vershik diagrams. In quantum field theory, cylindrical σ-algebras and associated measures on configuration spaces support renormalization procedures, as seen in bosonic QFTs where cylindrical Wigner measures capture semiclassical limits and local properties of field configurations, ensuring consistency across scales in interacting theories.

References

[1]
[PDF] 1 Random Vectors and Product Spaces
Definition 31 (Projection, Cylinder Set, and Product Measure). Let T be a set. For each t ∈ T, let (Xt,Ft) be a measurable space. Let X = Q t∈T. Xt be the ...
[2]
[PDF] Notes on Introductory Point-Set Topology - Cornell Mathematics
A first guess for how to define a topology on an infinite product Qi Xi would be to do the same thing as for finite products, taking as a basis the products U1.
[3]
An Introduction To Abstract Harmonic Analysis : Loomis,Lynn H.
Nov 12, 2006 · An Introduction To Abstract Harmonic Analysis. by: Loomis,Lynn H. Publication date: 1953/00/00. Topics: NATURAL SCIENCES, Mathematics, Analysis.Missing: sets | Show results with:sets
[4]
[PDF] Section 19. The Product Topology
Dec 6, 2016 · In this section we consider arbitrary products of topological spaces and give two topologies on these spaces, the box topology and the product.
[5]
[PDF] spl13.tex Lecture 13. 8.11.2010 8. The Borel-Cantelli lemmas and ...
Nov 8, 2010 · cylinder sets. The resulting probability space is called ... n=1(Ωn,An,µn). Example: Infinite coin tossing and the uniform distribution.
[6]
[PDF] Kolmogorov's theorem - Math@LSU
Here we shall consider the most basic form of Kolmogorov's theorem. It says that if we are given a family of probability measures µn on Rn, for all n ≥ 1,.
[7]
[PDF] Lecture Notes on Probability Theory - Jason Swanson
Nov 5, 2024 · cylinder sets, then B is a basis for a topology on X. The topology generated by B is called the product topology (see [10, Section 19]).
[8]
[PDF] MTH 304: General Topology
is open in the box topology, but not in the product topology. Proof. Consider 0 ∈ U. If U ∈ τp, then there must be a basic open set B such that 0 ∈ B and ...
[9]
[PDF] 2.6 Shift spaces and coding - University of Bristol
N invariant under the shift map σ) are metric spaces. The following sets, called cylinders, play an essential role in the study of shift spaces: Definition 2.7.
[10]
[PDF] notes on symbolic dynamics.
are called cylinder sets. In the ... Are there open sets in product topology that are not closed? Shift spaces with product topology are metrizable.
[11]
[PDF] 3. Shifts of finite type
shift spaces. These are the cylinder sets and are formed by fixing a finite set of co-ordinates. More precisely, in ΣA we define. [y−m,...,y−1,y0,y1,...,yn] ...
[12]
[PDF] generalized functions
This book, part of a series on "Generalized Functions," covers linear topological spaces and harmonic analysis in Euclidean and infinite-dimensional spaces.
[13]
[PDF] Levy processes and Levy white noise as tempered distributions
of tempered (Schwartz) distributions [13, 27]. It is natural to ask whether ... The σ-field generated by all cylinder sets is called the cylinder σ-algebra.
[14]
[PDF] A note on the σ-algebra of cylinder sets and all that - CCM
In this note we introduced and describe the different kinds of σ-algebras in infinite dimensional spaces. We emphasize the case of Hilbert spaces and nuclear ...<|control11|><|separator|>
[15]
[PDF] RANDOM ELEMENTS IN LINEAR SPACES
Let (E, X) be a separated, locally convex space. A cylinder set measure P on E is called a Gaussian measure if the projective system {PG} consists of Gaussian ...
[16]
product topological space in nLab
Aug 18, 2025 · The resulting product topological space is the category theoretic product of the original space in the category Top of topological spaces.
[17]
[PDF] Chapter 6 Products and Quotients
box topology definition of the product topology is the “right” definition to use. (Of course, the box topology and the product topology coincide for products.
[18]
[PDF] Chapter 4: Topological Spaces - UC Davis Mathematics
Definition 4.1 A topology on a nonempty set X is a collection of subsets of X, called open sets, such that: (a) the empty set 0 and the set X are open;. (b) the ...
[19]
[PDF] First Examples and General Properties of Subshifts
The cylinder sets form a basis of the product topology on Σ, i.e., a set is open in the product topology precisely if it can be written as arbitrary unions of ...<|control11|><|separator|>
[20]
[PDF] Theory of Probability - University of Texas at Austin
Definition 1.32 (Cylinder sets) Let {(Sγ,Sγ)}γ∈Γ be a family of measurable spaces, and let. (Qγ∈Γ. Sγ,⊗γ∈ΓSγ) be its product. A subset C ⊆ Qγ∈Γ. Sγ is ...
[21]
https://www.math.tau.ac.il/~peledron/Teaching/Brownian_motion/Sigma_algebra_notes.pdf
[22]
[PDF] Math 639: Lecture 1 - Measure theory background
Jan 24, 2017 · A π-system is a collection P of sets closed under finite intersections. A λ-system is a collection L of sets satisfying the following. Ω ∈ L.
[23]
[PDF] Note 10 Construction of general stochastic processes. The Daniell ...
is a 𝜋-system inside the algebra of cylinder sets. Both systems generate ℰI: any 𝜎-algebra which contains the special cylinder sets makes every projection ...
[24]
[PDF] COURSE NOTES ON MEASURE THEORY Week # 1 §1 Classes of ...
Cylinder sets in shift spaces. Let S be a finite set, and let X ∶= SN ... The set U ∈ B above is called the measure theoretic union of the hereditary ...
[25]
[PDF] PDF - 18.175 Theory of Probability
by algebra A, i.e. the minimal σ-algebra that contains all cylinders. Definition. A is called the cylindrical algebra and A is the cylindrical σ-algebra on RT .
[26]
[PDF] Product Measures and Extension Theorems
Nov 17, 2016 · Theorem 1. (Existence theorem for infinite product probability spaces) P on R extends uniquely to a countably additive probability measure on A.
[27]
[PDF] The Kolmogorov extension theorem - Jordan Bell
Jun 21, 2014 · An intersection of finitely many sets of the form A × MI\{t}, A ∈ Mt, is called a product cylinder. Lemma 5. The collection of all product ...
[28]
[PDF] Introduction to Ergodic theory
Oct 23, 2014 · A basis for the induced topology of this metric is given by cylinder sets. A cylinder set is a clopen subset of Ω determined by j ∈ Z, and a ...
[29]
[PDF] Ergodic Theory of Groups - Clara Löh - Universität Regensburg
Jun 3, 2020 · cylinder sets (Theorem A.1.13) shows that this action is ... where one only assumes the measure to be quasi-invariant under the equiva-.
[30]
ON ABSTRACT WIENER MEASURE* - Project Euclid
We denote by f^(G), the σ-algebra of cylinder sets [6, p. 66] ... Gross, Abstract Wiener spaces, Proceedings of the Fifth Berkeley Symposium, University of.