Space Of Measurable Functions Research Articles

On a new approach in the space of measurable functions

In this paper, we present a new modulus of continuity for locally integrable function spaces which is effected by the natural structure of the L_{p} space. After basic properties of it are expressed, we provide a quantitative type theorem for the rate of convergence of convolution type integral operators and iterates of them. Moreover, we state their global smoothness preservation property including the new modulus of continuity. Finally, the obtained results are performed to the Gauss-Weierstrass operators.

Optimal extensions of Lipschitz maps on metric spaces of measurable functions

We prove a factorization theorem for Lipschitz operators acting on certain subsets of metric spaces of measurable functions and with values on general metric spaces. Our results show how a Lipschitz operator can be extended to a subset of other metric space of measurable functions that satisfies the following optimality condition: it is the biggest metric space, formed by measurable functions, to which the operator can be extended preserving the Lipschitz constant. As an application, we show the coarsest metric that can be given for a metric space in which an order bounded lattice-valued-Lipschitz map is defined. Concrete examples involving the relevant space L0(μ) are given.

Banach spaces of GLT sequences and function spaces

Generalized locally Toeplitz (GLT) sequences of matrices originated from the spectral study of certain partial differential equations. To be more precise, such matrix sequences arise when we numerically approximate either partial differential equations or fractional differential equations using any discretization by local methods (finite differences, finite elements, finite volumes, isogeometric analysis, etc.). The study of the asymptotic spectral behavior of GLT sequences is important in analyzing the solution of the corresponding partial differential equations and in finding fast and efficient methods for the corresponding large linear systems. Approximating classes of sequences (a.c.s.) and spectral symbols are important notions connected to GLT sequences. Recently, G. Barbarino obtained some results regarding the theoretical aspects of such notions. He obtained the completeness of the space of matrix sequences with respect to pseudometric a.c.s. Also, he identified the space of GLT sequences with the space of measurable functions. In this article, we follow the same research line and obtain various connections between the subalgebras of matrix-sequence spaces and the subalgebras of function spaces. In some cases, these are identifications as Banach spaces and some of them are Banach algebra identifications. We also prove that the convergence notions in the sense of eigenvalue/singular value clustering are equivalent to the convergence with respect to the metrics introduced here. These convergence notions are related to the study of preconditioners in the case of matrix/operator sequences. As an application of our main results, we establish a Korovkin-type result in the setting of GLT sequences.

A Method for Studying Integral Equations by Using a Covering Set of the Nemytskii Operator in Spaces of Measurable Functions

A Method for Studying Integral Equations by Using a Covering Set of the Nemytskii Operator in Spaces of Measurable Functions

Orthogonally Biadditive Operators

In this article, we introduce and study a new class of operators defined on a Cartesian product of ideal spaces of measurable functions. We use the general approach of the theory of vector lattices. We say that an operator T : E × F ⟶ W defined on a Cartesian product of vector lattices E and F and taking values in a vector lattice W is orthogonally biadditive if all partial operators T y : E ⟶ W and T x : F ⟶ W are orthogonally additive. In the first part of the article, we prove that, under some mild conditions, a vector space of all regular orthogonally biadditive operators O B A r E , F ; W is a Dedekind complete vector lattice. We show that the set of all horizontally-to-order continuous regular orthogonally biadditive operators is a projection band in O B A r E , F ; W . In the last section of the paper, we investigate orthogonally biadditive operators on a Cartesian product of ideal spaces of measurable functions. We show that an integral Uryson operator which depends on two functional variables is orthogonally biadditive and obtain a criterion of the regularity of an orthogonally biadditive Uryson operator.

Non-convex functionals penalizing simultaneous oscillations along independent directions: rigidity estimates

We study a family of non-convex functionals $\{\mathcal{E}\}$ on the space of measurable functions$u: \Omega_1\times\Omega_2 \subset \mathbb{R}^{n_1}\times\mathbb{R}^{n_2} \to \mathbb{R}$. These functionals vanish on the non-convex subset $S(\Omega_1\times\Omega_2)$ formed by functions of the form $u(x_1,x_2)=u_1(x_1)$ or $u(x_1,x_2)=u_2(x_2)$. We investigate under which conditions the converse implication $\mathcal{E}(u) = 0 \Rightarrow u \in S(\Omega_1\times\Omega_2)$ holds. In particular, we show that the answer depends strongly on the smoothness of u. We also obtain quantitative versions of this implication by proving that (at least for some parameters) $\mathcal{E}(u)$ controls in a strong sense the distance of $u$ to $S(\Omega_1\times\Omega_2)$.

The topology on the space of measurable functions that is compatible with convergence in nonadditive measure

In this paper, given a nonadditive measure μ, a topology, which is compatible with convergence in μ-measure, is defined on the real linear space of all (equivalence classes of) measurable functions by using the distance introduced by Dunford and Schwartz. Some fundamental properties of the topology, such as the open sphere condition, the Hausdorff separation axiom, separability, approximation by continuous functions, linearlization, and metrizability, are related to the characteristics of a nonadditive measure.

Об устойчивости решений интегральных уравнений в классе измеримых функций

Consider the equation G(x)=(y,) ̃ where the mapping G acts from a metric space X into a space Y, on which a distance is defined, y ̃ ∈ Y. The metric in X and the distance in Y can take on the value ∞, the distance satisfies only one property of a metric: the distance between y,z ∈Y is zero if and only if y= z. For mappings X → Y the notions of sets of covering, Lipschitz property, and closedness are defined. In these terms, the assertion is obtained about the stability in the metric space X of solutions of the considered equation to changes of the mapping G and the element y ̃. This assertion is applied to the study of the integral equation f(t,∫_0^1▒K (t,s)x(s)ds,x(t))= y ̃(t),t ∈[0,1], with respect to an unknown Lebesgue measurable function x: [0,1] ∈R. Sufficient conditions are obtained for the stability of solutions (in the space of measurable functions with the topology of uniform convergence) to changes of the functions f,K,(y.) ̃

Atomic Operators in Vector Lattices

In this paper, we introduce a new class of operators on vector lattices. We say that a linear or nonlinear operator T from a vector lattice E to a vector lattice F is atomic if there exists a Boolean homomorphism Phi from the Boolean algebra {mathfrak {B}}(E) of all order projections on E to {mathfrak {B}}(F) such that Tpi =Phi (pi )T for every order projection pi in {mathfrak {B}}(E). We show that the set of all atomic operators defined on a vector lattice E with the principal projection property and taking values in a Dedekind complete vector lattice F is a band in the vector lattice of all regular orthogonally additive operators from E to F. We give the formula for the order projection onto this band, and we obtain an analytic representation for atomic operators between spaces of measurable functions. Finally, we consider the procedure of the extension of an atomic map from a lateral ideal to the whole space.

Some Properties Related With L^0 (Ω,F,μ) Space

The purpose of this paper is to investigate elementary properties of these measure and relation between these measure and we define a metric functions on the space of measurable functions and defined on finite measure space and we call the topology induced by p the topology of convergence in measure and we investigate now the connection between the convergence in metric of X and convergence in measure .
 

A simplified approach to subjective expected utility

I provide a novel simplified approach to Savage’s theory of subjective expected utility. Such an approach is based on abstract integral representation theorems in the space of measurable functions. The advantage of such an approach is that these results can be used to easily obtain variations on Savage’s theorem, such as representations with state-dependent utility or probability measures that can have atoms. Finally, I discuss how such an approach can be used in other settings such as decision making under ambiguity.

Log-optimal and rapid paths in von Neumann-Gale dynamical systems

Von Neumann-Gale dynamical systems are defined in terms of multivalued operators in spaces of random vectors, possessing certain properties of convexity and homogeneity. A central role in the theory of such systems is played by a special class of paths (trajectories) called rapid: they grow over each time period [t−1,t] in a sense faster than others. The paper establishes existence and characterization theorems for such paths showing, in particular, that any trajectory maximizing a logarithmic functional over a finite time horizon is rapid. The proof of this result is based on the methods of convex analysis in spaces of measurable functions. The study is motivated by the applications of the theory of von Neumann-Gale dynamical systems to the modeling of capital growth in financial markets with friction.

Bilinear Multipliers on Banach Function Spaces

Let X1,X2,X3 be Banach spaces of measurable functions in L0(R) and let m(ξ,η) be a locally integrable function in R2. We say that m∈BM(X1,X2,X3)(R) if Bm(f,g)(x)=∫R∫Rf^(ξ)g^(η)m(ξ,η)e2πi<ξ+η,x>dξdη, defined for f and g with compactly supported Fourier transform, extends to a bounded bilinear operator from X1×X2 to X3. In this paper we investigate some properties of the class BM(X1,X2,X3)(R) for general spaces which are invariant under translation, modulation, and dilation, analyzing also the particular case of r.i. Banach function spaces. We shall give some examples in this class and some procedures to generate new bilinear multipliers. We shall focus on the case m(ξ,η)=M(ξ-η) and find conditions for these classes to contain nonzero multipliers in terms of the Boyd indices for the spaces.

Distance of convergence in measure on spaces of measurable functions

In this paper we investigate the distance of convergence in measure whenever the measure is not $$\sigma $$ -finite and identify the topological coreflection of this approach structure with a direct proof. Also we investigate its connections with metric approach structures on spaces of measurable functions and obtain some results for integrability of measurable functions in this setting.

Locally Convex Limit Spaces of Measurable Functions with Order Units and Its Duals

We consider linear normed spaces of measurable functions dominated by positive measurable function powered by real positive parameter. Also, we consider its dual and predual, and we propose a method for constructing a limit spaces of these functional spaces taken by power parameter. We prove that these limit spaces are (LF)-spaces and also prove that the limit spaces presume the relation of duality, i.e., the limit space of predual spaces is predual for the limit space of dominated functions, and the limit space of duals is dual for it. Also, the limit space of predual spaces is embedded into the limit space of dual spaces.

Best approximation properties in spaces of measurable functions

AbstractWe research proximinality of μ‐sequentially compact sets and μ‐compact sets in measurable function spaces. Next we show a correspondence between the Kadec–Klee property for convergence in measure and μ‐compactness of the sets in Banach function spaces. Also the property S is investigated in Fréchet spaces and employed to provide the Kadec–Klee property for local convergence in measure. We discuss complete criteria for continuity of metric projection in Fréchet spaces with respect to the Hausdorff distance. Finally, we present the necessary and sufficient condition for continuous metric selection onto a one‐dimensional subspace in sequence Lorentz spaces .

Equivalence between GLT sequences and measurable functions

The theory of Generalized Locally Toeplitz (GLT) sequences of matrices has been developed in order to study the asymptotic behaviour of particular spectral distributions when the dimension of the matrices tends to infinity. Key concepts in this theory are the notions of Approximating Classes of Sequences (a.c.s.) and spectral symbols that lead to defining a metric structure on the space of matrix sequences and provide a link with the measurable functions. In this paper we prove additional results regarding theoretical aspects, such as the completeness of the matrix sequences space with respect to the metric a.c.s. and the identification of the space of GLT sequences with the space of measurable functions.

Variation of Parameters for Causal Operator Differential Equations

The operator T from a domain D into the space of measurable functions is called a nonanticipating (causal) operator if the past information is independent from the future outputs. We will study the solution x(t) of a nonlinear operator differential equation where its changes depends on the causal operator T, and semigroup of operator A(t), and all initial parameters (t0, x0) . The initial information is described x(t)=φ(t) for almost all t ≤t0 and φ(t0) =φ0. We will study the nonlinear variation of parameters (NVP) for this type of nonanticipating operator differential equations and develop Alekseev type of NVP.

Computability on measurable functions

For a computable measure space with computable σ -finite measure we study computability on the space of measurable functions. First we prove that for a natural multi-representation the measurable sets are closed under finite union and intersection and under countable union. We introduce a natural multi-representation of the measurable functions to a computable topological space and prove that composition with continuous functions is computable. Canonically, for a computable metric space the associated computable topological space has a basis of balls B(a, s) with center from the dense set and rational radius. From a given sequence (μk)k of finite Borel measures we can compute a dense sequence (rj )j of radii such that μk(S) = 0f o ra l lk and all spheres S = S(a,rj ) with a from the dense set and j ∈ N. For measurable functions to computable compact computable metric spaces the natural multi-representation is equivalent to a multi-representation via fast uniformly converging sequences of simple functions. For non-negative measurable numerical functions the sequences can be chosen to be non-decreasing. Some results on integration are added.

Inductive and projective limits of Banach spaces of measurable functions with order unities with respect to power parameter

We prove that a measurable function f is bounded and invertible if and only if there exist at least two equivalent norms by order unit spaces with order unities fα and fβ with α > β > 0. We show that it is natural to understand the limit of ordered vector spaces with order unities fα (α approaches to infinity) as a direct sum of one inductive and one projective limits. We also obtain some properties for the corresponding limit topologies.