Sequential closures of $\sigma $-subalgebras for a vector measure

Werner J. Ricker

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Displaying similar documents to “Sequential closures of $σ$ -subalgebras for a vector measure”

A compact Hausdorff topology that is a T₁-complement of itself

Dmitri Shakhmatov, Michael Tkachenko (2002)

Fundamenta Mathematicae

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Topologies τ₁ and τ₂ on a set X are called T₁-complementary if τ₁ ∩ τ₂ = X∖F: F ⊆ X is finite ∪ ∅ and τ₁∪τ₂ is a subbase for the discrete topology on X. Topological spaces $(X, τ_{X})$ and $(Y, τ_{Y})$ are called T₁-complementary provided that there exists a bijection f: X → Y such that $τ_{X}$ and $f^{- 1} (U) : U \in τ_{Y}$ are T₁-complementary topologies on X. We provide an example of a compact Hausdorff space of size $2^{}$ which is T₁-complementary to itself ( denotes the cardinality of the continuum). We prove that the existence of a compact...

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

S. Gabriyelyan, J. Kąkol, G. Plebanek (2016)

Studia Mathematica

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Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of $C_{k} (X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_{ℝ}$ -space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space $C_{k} (X)$ is Ascoli iff $C_{k} (X)$ is a $k_{ℝ}$ -space iff X is locally compact. Moreover, $C_{k} (X)$ endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability...

Topological properties of some spaces of continuous operators

Marian Nowak (2016)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let $C_{b} (X, E)$ be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study topological properties of the space $L_{β} (C_{b} (X, E), F)$ of all $(β, | | \cdot | |_{F})$ -continuous linear operators from $C_{b} (X, E)$ to F, equipped with the topology $τ_{s}$ of simple convergence. If X is a locally compact paracompact space (resp. a P-space), we characterize $τ_{s}$ -compact subsets of $L_{β} (C_{b} (X, E), F)$ in terms of properties of the corresponding sets of the representing...

On FU( $p$ )-spaces and $p$ -sequential spaces

Salvador García-Ferreira (1991)

Commentationes Mathematicae Universitatis Carolinae

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Following Kombarov we say that $X$ is $p$ -sequential, for $p \in α^{*}$ , if for every non-closed subset $A$ of $X$ there is $f \in^{α} X$ such that $f (α) \subseteq A$ and $\bar{f} (p) \in X ∖ A$ . This suggests the following definition due to Comfort and Savchenko, independently: $X$ is a FU( $p$ )-space if for every $A \subseteq X$ and every $x \in A^{-}$ there is a function $f \in^{α} A$ such that $\bar{f} (p) = x$ . It is not hard to see that $p \leq_{RK} q$ ( $\leq_{RK}$ denotes the Rudin–Keisler order) $\Leftrightarrow$ every $p$ -sequential space is $q$ -sequential $\Leftrightarrow$ every FU( $p$ )-space is a FU( $q$ )-space. We generalize the spaces $S_{n}$ to construct examples of...

Invariant subspaces for operators in a general II1-factor

Uffe Haagerup, Hanne Schultz (2009)

Publications Mathématiques de l'IHÉS

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Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace $𝒦 = 𝒦_{T} (B)$ affiliated with ℳ, such that the Brown measure of ${T |}_{𝒦}$ is concentrated...

Lattice copies of c₀ and $ℓ^{\infty}$ in spaces of integrable functions for a vector measure

S. Okada, W. J. Ricker, E. A. Sánchez Pérez

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The spaces L¹(m) of all m-integrable (resp. $L ¹_{w} (m)$ of all scalarly m-integrable) functions for a vector measure m, taking values in a complex locally convex Hausdorff space X (briefly, lcHs), are themselves lcHs for the mean convergence topology. Additionally, $L ¹_{w} (m)$ is always a complex vector lattice; this is not necessarily so for L¹(m). To identify precisely when L¹(m) is also a complex vector lattice is one of our central aims. Whenever X is sequentially complete, then this is the case. If,...

Order intervals in $C (K)$ . Compactness, coincidence of topologies, metrizability

Zbigniew Lipecki (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let $K$ be a compact space and let $C (K)$ be the Banach lattice of real-valued continuous functions on $K$ . We establish eleven conditions equivalent to the strong compactness of the order interval $[0, x]$ in $C (K)$ , including the following ones: (i) ${x > 0}$ consists of isolated points of $K$ ; (ii) $[0, x]$ is pointwise compact; (iii) $[0, x]$ is weakly compact; (iv) the strong topology and that of pointwise convergence coincide on $[0, x]$ ; (v) the strong and weak topologies coincide on $[0, x]$ . Moreover, the weak topology and that of pointwise...

Characteristic points, rectifiability and perimeter measure on stratified groups

Valentino Magnani (2006)

Journal of the European Mathematical Society

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We establish an explicit connection between the perimeter measure of an open set $E$ with $C^{1}$ boundary and the spherical Hausdorff measure $S^{Q - 1}$ restricted to $\partial E$ , when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and $Q$ denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of $E$ is less than or equal to $S^{Q - 1} (\partial E)$ up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli,...

Kempisty's theorem for the integral product quasicontinuity

Zbigniew Grande (2006)

Colloquium Mathematicae

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A function f: ℝⁿ → ℝ satisfies the condition $Q_{i} (x)$ (resp. $Q_{s} (x)$ , $Q_{o} (x)$ ) at a point x if for each real r > 0 and for each set U ∋ x open in the Euclidean topology of ℝⁿ (resp. strong density topology, ordinary density topology) there is an open set I such that I ∩ U ≠ ∅ and $| (1 / μ (U \cap I)) \int_{U \cap I} f (t) d t - f (x) | < r$ . Kempisty’s theorem concerning the product quasicontinuity is investigated for the above notions.

On the structure of non-dentable subsets of $C (ω^{ω^{k}})$

Pericles D. Pavlakos, Minos Petrakis (2011)

Studia Mathematica

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It is shown that there is no closed convex bounded non-dentable subset K of $C (ω^{ω^{k}})$ such that on subsets of K the PCP and the RNP are equivalent properties. Then applying the Schachermayer-Rosenthal theorem, we conclude that every non-dentable K contains a non-dentable subset L so that on L the weak topology coincides with the norm topology. It follows from known results that the RNP and the KMP are equivalent on subsets of $C (ω^{ω^{k}})$ .

Denseness and Borel complexity of some sets of vector measures

Zbigniew Lipecki (2004)

Studia Mathematica

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Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets $_{ν} (X)$ and $_{ν} (X)$ of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient...

Radon-Nikodym property

Surjit Singh Khurana (2017)

Commentationes Mathematicae Universitatis Carolinae

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For a Banach space $E$ and a probability space $(X, 𝒜, λ)$ , a new proof is given that a measure $μ : 𝒜 \to E$ , with $μ ≪ λ$ , has RN derivative with respect to $λ$ iff there is a compact or a weakly compact $C \subset E$ such that ${| μ |}_{C} : 𝒜 \to [0, \infty]$ is a finite valued countably additive measure. Here we define ${| μ |}_{C} (A) = sup {\sum_{k} | 〈 μ (A_{k}), f_{k} 〉 |}$ where ${A_{k}}$ is a finite disjoint collection of elements from $𝒜$ , each contained in $A$ , and ${f_{k}} \subset E^{'}$ satisfies ${sup}_{k} | f_{k} (C) | \leq 1$ . Then the result is extended to the case when $E$ is a Frechet space.

Displaying similar documents to “Sequential closures of σ -subalgebras for a vector measure”

Displaying similar documents to “Sequential closures of $σ$ -subalgebras for a vector measure”