Let $X$ and $Y$ be Banach spaces and $(Ω,Σ,μ)$ a finite measure space. In this note we introduce the space $L^p[μ;L(X,Y)]$ consisting of all (equivalence classes of) functions $Φ:Ω\mapsto L(X,Y)$ such that $ω\mapsto Φ(ω)x$ is strongly $μ$-measurable for all $x\in X$ and $ω\mapsto Φ(ω)f(ω)$ belongs to $L^1(μ;Y)$ for all $f\in L^{p'}(μ;X)$, $1/p+1/p'=1$. We show that functions in $L^p[μ;Ł(X,Y)]$ define operator-valued measures with bounded $p$-variation and use these spaces to obtain an isometric characterization of the space of all $L(X,Y)$-valued multipliers acting boundedly from $L^p(μ;X)$ into $L^q(μ;Y)$, $1\le q< p<\infty$.
In this work the problem about an existence of non-measurable automorphisms of Lie groups finite and as well infinite dimensional over the field of real numbers and also over the non-archimedean local fields is investigated. Non-measurability of automorphisms is considered relative to real-valued measures and also measures with values in non-archimedean local fields. Their existence is proved and a procedure for their construction is given. Their application for a construction of non-measurable irreducible unitary representations is demonstrated.
For a measurable space ($X,\mathcal{A}$), let $\mathcal{M}(X,\mathcal{A})$ be the corresponding ring of all real valued measurable functions and let $μ$ be a measure on ($X,\mathcal{A}$). In this paper, we generalize the so-called $m_μ$ and $U_μ$ topologies on $\mathcal{M}(X,\mathcal{A})$ via an ideal $I$ in the ring $\mathcal{M}(X,\mathcal{A})$. The generalized versions will be referred to as the $m_{μ_{I}}$ and $U_{μ_{I}}$ topology, respectively, throughout the paper. $L_{I}^{\infty} \left(μ\right)$ stands for the subring of $\mathcal{M}(X,\mathcal{A})$ consisting of all functions that are e…
We extend the classical Mercer theorem to reproducing kernel Hilbert spaces whose elements are functions from a measurable space $X$into $\mathbb C^n$. Given a finite measure $μ$ on $X$, we represent the reproducing kernel $K$ as convergent series in terms of the eigenfunctions of a suitable compact operator depending on $K$ and $μ$. Our result holds under the mild assumption that $K$ is measurable and the associated Hilbert space is separable. Furthermore, we show that $X$ has a natural second countable topology with respect to which the eigenfunctions are continuous and the series representi…
In this paper, we define the Dunford-Henstock-Kurzweil and the Dunford-McShane integrals of Banach space valued functions defined on a bounded Lebesgue measurable subset of m-dimensional Euclidean space Rm. We will show that the new integrals are "natural" extensions of the McShane and the Henstock-Kurzweil integrals from m-dimensional closed non-degenerate intervals to m-dimensional bounded Lebesgue measurable sets. As applications, we will present full descriptive characterizations of the McShane and Henstock-Kurzweil integrals in terms of our integrals. Moreover, a relationship between new …
We focus on measurability and integrability for set valued functions in non-necessarily separable Fréchet spaces. We prove some properties concerning the equivalence between different classes of measurable multifunctions. We also provide useful characterizations of Pettis set-valued integrability in the announced framework. Finally, we indicate applications to Volterra integral inclusions.
We define a p-norm in the context of quantum random variables, measurable operator-valued functions with respect to a positive operator-valued measure. This norm leads to a operator-valued L^p space that is shown to be complete. Various other norm candidates are considered as well as generalizations of Hölder's inequality to this new context.
The Egoroff theorem for measurable $\bold X$-valued functions and operator-valued measures $\bold m: Σ\to L(\bold X, \bold Y)$, where $Σ$ is a $σ$-algebra of subsets of $T \neq \emptyset$ and $\bold X$, $\bold Y$ are both locally convex spaces, is proved. The measure is supposed to be atomic and the convergence of functions is net.
In the present paper we investigate $L_0$-valued states and Markov operators on $ C^*$-algebras over $L_0$. In particular, we give representations for $L_0$-valued state and Markov operators on $ C^*$ algebras over $L_0$, respectively, as measurable bundles of states and Markov operators. Moreover, we apply the obtained representations to study certain ergodic properties of $ C^*$-dynamical systems over $L_0$.
In the present paper we investigate Banach--Kantorovich algebras over faithful solid subalgebras of algebras measurable functions. We prove that any Banach--Kantorovich algebra over faithful solid subalgebras of algebra measurable functions represented as a measurable bundle of Banach algebras with vector-valued lifting. We apply such representation to the spectrum of elements Banach--Kantorovich algebras.
We introduce a real-valued measure ${m_L}$ on non-Archimedean ordered fields $(\mathbb{F},<)$ that extend the field of real numbers $(\mathbb{R},<)$. The definition of ${m_L}$ is inspired by the Loeb measures of hyperreal fields in the framework of Robinson's analysis with infinitesimals. The real-valued measure ${m_L}$ turns out to be general enough to obtain a canonical measurable representative in $\mathbb{F}$ for every Lebesgue measurable subset of $\mathbb{R}$, moreover, the measure of the two sets is equal. In addition, $m_L$ it is more expressive than a class of non-Archimedean uniform …
A Daniell-Stone type characterization theorem for Aumann integrals of set-valued measurable functions will be proven. It is assumed that the values of these functions are closed convex upper sets, a structure that has been used in some recent developments in set-valued variational analysis and set optimization. It is shown that the Aumann integral of such a function is also a closed convex upper set. The main theorem characterizes the conditions under which a functional that maps from a certain collection of measurable set-valued functions into the set of all closed convex upper sets can be wr…
We prove a version of the Lebesgue Differentiation Theorem for mappings that are defined on a measure space and take values into a metric space, with respect to the differentiation basis induced by a von Neumann lifting. As a consequence, we obtain a lifting theorem for the space of sections of a measurable Banach bundle and a disintegration theorem for vector measures whose target is a Banach space with the Radon-Nikodým property.
Let $(S,\cdot)$ be a semigroup and $\mathfrak{m}$ be a $σ$-algebra on $S$. We say $(S,\cdot,\mathfrak{m})$ is a measurable semigroup if $π:S\times S\longrightarrow S$ by $π(x,y)=x\cdot y$ is a measurable function. In this paper , we consider to $\mathfrak{m}^β$ as the collection of all ultrafilters on $\mathfrak{m}$. We show that $\mathfrak{m}^β$ is a compact right topological semigroup respect to generated topology by $σ$-algebra $\mathfrak{m}$ on $\mathfrak{m}^β$. Also we study some elementary properties of the algebraic structure of $\mathfrak{m}^β$.
The abscissas of convergence, uniform convergence and absolute convergence of vector valued Dirichlet series with respect to the original topology and with respect to the weak topology $σ(X,X')$ of a locally convex space $X$, in particular of a Banach space $X$, are compared. The relation of their coincidence with geometric or topological properties of the underlying space $X$ is investigated. Cotype in the context of Banach spaces, and nuclearity and certain topological invariants for Fréchet spaces play a relevant role.
Let $\mathcal{M}(Ω, μ)$ denote the algebra of all scalar-valued measurable functions on a measure space $(Ω, μ)$. Let $B \subset \mathcal{M}(Ω, μ)$ be a set of finitely supported measurable functions such that the essential range of each $f \in B$ is a subset of $\{ 0,1 \}$. The main result of this paper shows that for any $p \in (0, \infty)$, $B$ has strict $p$-negative type when viewed as a metric subspace of $L_{p}(Ω, μ)$ if and only if $B$ is an affinely independent subset of $\mathcal{M}(Ω, μ)$ (when $\mathcal{M}(Ω, μ)$ is considered as a real vector space). It follows that every two-valu…
We investigate some properties and convergence theorem of Kluvánek-Lewis-Henstock $\m-$integrability for $\m-$measurable functions that we introduced in \cite{ABH}. We give a $\m-$a.e. convergence version of Dominated (resp. Bounded) Convergence Theorem for $\m.$ We introduce Kluvánek-Lewis-Henstock integrable of scalar-valued functions with respect to a set valued measure in a Banach space. Finally we introduce $(KL)-$type Dominated Convergence Theorem for the set-valued Kluvánek-Lewis-Henstock integral.
We define a monad M on a category of measurable bornological sets, and we show how this monad gives rise to a theory of vector-valued integration that is related to the notion of Pettis integral. We show that an algebra X of this monad is a bornological locally convex vector space endowed with operations which associate vectors \int f dm in X to incoming maps f:T --> X and measures m on T. We prove that a Banach space is an M-algebra as soon as it has a Pettis integral for each incoming bounded weakly-measurable function. It follows that all separable Banach spaces, and all reflexive Banach sp…
We discuss Lebesgue spaces $\mathcal{L}^p([a,b],E)$ of Lusin measurable vector-valued functions and the corresponding vector spaces $AC_{L^p}([a,b],E)$ of absolutely continuous functions. These can be used to construct Lie groups $AC_{L^p}([a,b],G)$ of absolutely continuous functions with values in an infinite-dimensional Lie group $G$. We extend the notion of $L^p$-regularity of infinite-dimensional Lie groups introduced by Glöckner to this setting and adopt several results and tools.
We derive two types of representation results for increasing convex functionals in terms of countably additive measures. The first is a max-representation of functionals defined on spaces of real-valued continuous functions and the second a sup-representation of functionals defined on spaces of real-valued Borel measurable functions. Our assumptions consist of sequential semicontinuity conditions which are easy to verify in different applications.
We show a Dvoretsky-Rogers type Theorem for the adapted version of the $q$-summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guaranty that the space has to be finite dimensional, contrarily to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, sinc…
If $μ_1,μ_2,\dots$ are positive measures on a measurable space $(X,Σ)$ and $v_1,v_2, \dots$ are elements of a Banach space ${\mathbb E}$ such that $\sum_{n=1}^\infty \|v_n\| μ_n(X) < \infty$, then $ω(S)= \sum_{n=1}^\infty v_n μ_n(S)$ defines a vector measure of bounded variation on $(X,Σ)$. We show ${\mathbb E}$ has the Radon-Nikodym property if and only if every ${\mathbb E}$-valued measure of bounded variation on $(X,Σ)$ is of this form. As an application of this result we show that under natural conditions an operator defined on positive measures, has a unique extension to an operator defin…
In this paper, we fully characterize the duality mapping over the space of matrices that are equipped with Schatten norms. Our approach is based on the analysis of the saturation of the Hölder inequality for Schatten norms. We prove in our main result that, for $p\in (1,\infty)$, the duality mapping over the space of real-valued matrices with Schatten-$p$ norm is a continuous and single-valued function and provide an explicit form for its computation. For the special case $p = 1$, the mapping is set-valued; by adding a rank constraint, we show that it can be reduced to a Borel-measurable singl…
The set of all maximal ideals of the ring $\mathcal{M}(X,\mathcal{A})$ of real valued measurable functions on a measurable space $(X,\mathcal{A})$ equipped with the hull-kernel topology is shown to be homeomorphic to the set $\hat{X}$ of all ultrafilters of measurable sets on $X$ with the Stone-topology. This yields a complete description of the maximal ideals of $\mathcal{M}(X,\mathcal{A})$ in terms of the points of $\hat{X}$. It is further shown that the structure spaces of all the intermediate subrings of $\mathcal{M}(X,\mathcal{A})$ containing the bounded measurable functions are one and t…
There are two definitions of the measurable functional on the topological vector space: as a linear and measurable real-valued function and as a pointwise limit of the sequence of the continious linear functionals. In general case they are not equivalent, but in some cases it is so, for example, in the case of gaussian measures. There is one natural generalization of the gaussian measures - the convex measures. In this paper this equivalence was proved for the some classes of convex measures.
We show that if $G$ is an amenable topological group, then the topological group $L^{0}(G)$ of strongly measurable maps from $([0,1],λ)$ into $G$ endowed with the topology of convergence in measure is whirly amenable, hence extremely amenable. Conversely, we prove that a topological group $G$ is amenable if $L^{0}(G)$ is.
We study the Lusin approximation problem for real-valued measurable functions on Carnot groups. We prove that k-approximate differentiability almost everywhere is equivalent to admitting a Lusin approximation by $C^{k}_{\mathbb{G}}$ maps. We also prove that existence of an approximate (k-1)-Taylor polynomial almost everywhere is equivalent to admitting a Lusin approximation by maps in a suitable Lipschitz function space.
We introduce a vector-valued version of a uniform algebra, called the vector-valued function space over a uniform algebra. The diameter two properties of the vector-valued function space over a uniform algebra on an infinite compact Hausdorff space are investigated. Every nonempty relatively weakly open subset of the unit ball of a vector-valued function space $A(K, (X, τ))$ over an infinite dimensional uniform algebra has the diameter two, where $τ$ is a locally convex Hausdorff topology on a Banach space $X$ compatible to a dual pair. Under the assumption on $X$ being uniformly convex with n…
Consider a Banach space valued measurable function $f$ and an operator $u$ from the space where {$f$} takes values. If $f $ is Pettis integrable, a classical result due to J. Diestel shows that composing it with $u$ gives a Bochner integrable function $u \circ f$ whenever $u$ is absolutely summing. In a previous work we have shown that a well-known interpolation technique for operator ideals allows to prove under some requirements that a composition of a $p$-Pettis integrable function with a $q$-summing operator provides an $r$-Bochner integrable function. In this paper a new abstract inclusio…
We establish a Fenchel-Moreau type theorem for proper convex functions $f\colon X\to \bar{L}^0$, where $(X, Y, \langle \cdot,\cdot \rangle)$ is a dual pair of Banach spaces and $\bar L^0$ is the space of all extended real-valued functions on a $σ$-finite measure space. We introduce the concept of stable lower semi-continuity which is shown to be equivalent to the existence of a dual representation $$\smash{ f(x)=\sup_{y \in L^0(Y)} \left\{\langle x, y \rangle - f^\ast(y)\right\}, \quad x\in X,} $$ where $L^0(Y)$ is the space of all strongly measurable functions with values in $Y$, and $\langle…
We generalize Lyapunov's convexity theorem for classical (scalar-valued) measures to quantum (operator-valued) measures. In particular, we show that the range of a nonatomic quantum probability measure is a weak*-closed convex set of quantum effects (positive operators bounded above by the identity operator) under a sufficient condition on the non-injectivity of integration. To prove the operator-valued version of Lyapunov's theorem, we must first define the notions of essentially bounded, essential support, and essential range for quantum random variables (Borel measurable functions from a se…
We study the differential equation $\frac{\partial G}{\partial\bar z}=g$ with an unbounded Banach-valued Bochner measurable function $g$ on the open unit disk $\mathbb D\subset\mathbb C$. We prove that under some conditions on the growth and essential support of $g$ such equation has a bounded solution given by a continuous linear operator. The obtained results are applicable to the Banach-valued corona problem for the algebra of bounded holomorphic functions on $\mathbb D$ with values in a complex commutative unital Banach algebra.
In this {\color{red}{paper}} we discuss about the $ap-$Henstock-Kurzweil integrable functions on a topological vector spaces. Basic results of $ap-$Henstock-Kurzweil integrable functions are discussed here. We discuss the equivalence of the $ap-$Henstock-Kurzweil integral on a topological vector spaces and the vector valued $ap-$Henstock-Kurzweil integral. Finally, several convergence theorems are studied.
Control Measures for Bochner $L_{0}$-Valued Vector Measures
Faculty of Mathematics and Computer Science A. Mickiewicz University（波兹南亚当·密茨凯维奇大学数学与计算机科学学院）；Department of Statistics State University of Campinas (UNICAMP)（坎皮纳斯州立大学统计系）
In this article we have studied bicomplex valued measurable functions on an arbitrary measurable space. We have established the bicomplex version of Lebesgue's dominated convergence theorem and some other results related to this theorem. Also we have proved the bicomplex version of Lebesgue-Radon-Nikodym theorem. Finally we have introduced the idea of hyperbolic version of invariant measure.
We derive sufficient conditions for the surjectivity of the Cauchy-Riemann operator $\overline{\partial}$ between spaces of weighted smooth Fréchet-valued functions. This is done by establishing an analog of Hörmander's theorem on the solvability of the inhomogeneous Cauchy-Riemann equation in a space of smooth $\mathbb{C}$-valued functions whose topologyis given by a whole family of weights. Our proof relies on a weakened variant of weak reducibility of the corresponding subspace of holomorphic functions in combination with the Mittag-Leffler procedure. Using tensor products, we deduce the co…
Suppose $α$ is a rotationally symmetric norm on $L^{\infty}\left(\mathbb{T}\right) $ and $β$ is a "nice" norm on $L^{\infty}\left(Ω,μ\right) $ where $μ$ is a $σ$-finite measure on $Ω$. We prove a version of Beurling's invariant subspace theorem for the space $L^β\left(μ,H^α\right) .$ Our proof uses the recent version of Beurling's theorem on $H^α\left(\mathbb{T}\right) $ proved by the first author and measurable cross-section techniques. Our result significantly extends a result of H. Rezaei, S. Talebzadeh, and D. Y. Shin.
We study the Banach algebras ${\rm C}(X, R)$ of continuous functions from a compact Hausdorff topological space $X$ to a Banach ring $R$ whose topology is discrete. We prove that the Berkovich spectrum of ${\rm C}(X, R)$ is homeomorphic to $ζ(X) \times {\mathcal M}(R)$, where $ζ(X)$ is the Banaschewski compactification of $X$ and ${\mathcal M}(R)$ is the Berkovich spectrum of $R$. We study how the topology of the spectrum of ${\rm C}(X, R)$ is related to the notion of homotopy Zariski open embedding used in derived geometry. We find that the topology of $ζ(X)$ can be easily reconstructed from …
Let G be a locally compact group, and let U be its unitary representation on a Hilbert space H. Endow the space L(H) of linear bounded operators on H with weak operator topology. We prove that if U is a measurable map from G to L(H) then it is continuous. This result was known before for separable H. To prove this, we generalize a known theorem on nonmeasuralbe unions of point finite families of null sets. We prove also that the following statement is consistent with ZFC: every measurable homomorphism from a locally compact group into any topological group is continuous. This relies, in turn, …
Frank和Larson在有单位元的C *-代数上定义了Hilbert C *-模上的框架，arXiv.0707.3272v1研究了希尔伯特空间上的算子值框架。[math. FA].本文的目的是在sigma-单位C * 代数的Hilbert C *-模上引入算子值框架.定理1.4重新表述了Frank和Larson给出的关于一系列秩一算子在严格拓扑中收敛的定义。定理2.2.证明了算子值框架的框架变换和框架投影是乘子代数中一系列元素的严格拓扑的极限，因而属于乘子代数.定理3.3证明了两个算子值框架右相似当且仅当它们共享同一个框架投影.定理3.4建立了乘子代数中投影的Murray-von Neumann等价类与算子值框架的右相似等价类之间的一一对应关系，并给出了给定Hilbert C *-模上所有Parseval算子值框架的参数化。然后定义左相似性，命题3.9成立时，两个左酉等价框架也是右酉等价的。
In this paper we study maximal $L^p$-regularity for evolution equations with time-dependent operators $A$. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the $L^p$-boundedness of a class of vector-valued singular integrals which does not rely on Hörmander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of $m$-th order elliptic operators $A$ with time and space-dependent coefficients. Here the highest order c…
We investigate the possibility of replacing the topology of convergence in probability with convergence in $L^1$. A characterization of continuous linear functionals on the space of measurable functions is also obtained.
If G is a countable discrete group acting linearly on a finite-dimensional vector space over any topological field, then the groups of coboundaries are closed for the product topology in all degrees, and hence the cohomology is reduced in all degrees. This is deduced from a more general automatic-closure theorem for continuous linear transformations between inverse limits of finite-dimensional vector spaces.
