Real-valued Measurable Functions Research Articles

Sliding window method for measurable functions on neutrosophic normed spaces

We aim to introduce the nonnegative real-valued functions class and the research of Mursaleen and Mohiuddin. We introduce the sliding window idea approaches within the neutrosophic normed spaces range [0;∞) for real-valued measurable functions in order to accomplish aim. Additionally,using summability techniques, we derive a number of theorems and samples.

Some properties of algebras of real-valued measurable functions

Some properties of algebras of real-valued measurable functions

Gaussian estimates for heat kernels of higher order Schrödinger operators with potentials in generalized Schechter classes

Let m ∈ N $m\in \mathbb {N}$ , P ( D ) : = ∑ | α | = 2 m ( − 1 ) m a α D α $P(D):=\sum _{|\alpha |=2m}(-1)^m a_\alpha D^\alpha$ be a 2 m $2m$ -order homogeneous elliptic operator with real constant coefficients on R n $\mathbb {R}^n$ , and V $V$ a real-valued measurable function on R n $\mathbb {R}^n$ . In this article, the authors introduce a new generalized Schechter class concerning V $V$ and show that the higher order Schrödinger operator L : = P ( D ) + V $\mathcal {L}:=P(D)+V$ possesses a heat kernel that satisfies the Gaussian upper bound and the Hölder regularity when V $V$ belongs to this new class. The Davies–Gaffney estimates for the associated semigroup and their local versions are also given. These results pave the way for many further studies on the analysis of L $\mathcal {L}$ .

Generalizations of some concentration inequalities

For a real-valued measurable function f and a nonnegative, nondecreasing function ϕ, we first obtain a Chebyshev type inequality which provides an upper bound for ϕ(λ1)μ({x∈Ω:f(x)≥λ1})+∑k=2nϕ(λk)−ϕ(λk−1)μ({x∈Ω:f(x)≥λk}), where 0<λ1<λ2<⋯<λn<∞. Using this, generalizations of a few concentration inequalities such as Markov, reverse Markov, Bienaymé–Chebyshev, Cantelli and Hoeffding inequalities are obtained.

Monotonicity properties and solvability of dominated best approximation problem in Orlicz spaces equipped with s-norms

In the paper, Wisła (J Math Anal Appl 483(2):123659, 2020, 10.1016/j.jmaa.2019.123659), it was proved that the classical Orlicz norm, Luxemburg norm and (introduced in 2009) p-Amemiya norm are, in fact, special cases of the s-norms defined by the formula left| xright| _{Phi ,s}=inf _{k>0}frac{1}{k}sleft( int _T Phi (kx)dmu right) , where s and Phi are an outer and Orlicz function respectively and x is a measurable real-valued function over a sigma -finite measure space (T,Sigma ,mu ). In this paper the strict monotonicity, lower and upper uniform monotonicity and uniform monotonicity of Orlicz spaces equipped with the s-norm are studied. Criteria for these properties are given. In particular, it is proved that all of these monotonicity properties (except strict monotonicity) are equivalent, provided the outer function s is strictly increasing or the measure mu is atomless. Finally, some applications of the obtained results to the best dominated approximation problems are presented.

On the $\Delta _{{\Lambda ^2}}^f$-Statistical Convergence on Product Time Scale

In this paper, we first define a new density of a $\Delta $-measurable subset of a product time scale ${\Lambda ^2}$ with respect to an unbounded modulus function. Then, by using this definition, we introduce the concepts of $\Delta _{{\Lambda ^2}}^f$-statistical convergence and $\Delta _{{\Lambda ^2}}^f$-statistical Cauchy for a $\Delta $-measurable real-valued function defined on product time scale ${\Lambda ^2}$ and also obtain some results about these new concepts. Finally, we present the definition of strong $\Delta _{{\Lambda ^2}}^f$-Cesaro summability on ${\Lambda ^2}$ and investigate the connections between these new concepts.

Rearrangement and polarization

The paper has two main goals. The first is to take a new approach to rearrangements on certain classes of measurable real-valued functions on Rn. Rearrangements are maps that are monotonic (up to sets of measure zero) and equimeasurable, i.e., they preserve the measure of super-level sets of functions. All the principal known symmetrization processes for functions, such as Steiner and Schwarz symmetrization, are rearrangements, and these have a multitude of applications in diverse areas of the mathematical sciences. The second goal is to understand which properties of rearrangements characterize polarization, a special rearrangement that has proved particularly useful in a number of contexts. In order to achieve this, new results are obtained on the structure of measure-preserving maps on convex bodies and of rearrangements generally.

$$I_\Delta $$ and $$I^*_\Delta $$-convergence on time scales

In this paper, we introduce the concept of $$\Delta $$ -weak ideal on a time scale T and using it we introduce and study the notions of $$I_{\Delta }$$ -convergence and $$I^{*}_{\Delta }$$ -convergence of a $$\Delta $$ -measurable real-valued function f defined on T. The notion of $$I_{\Delta }$$ -convergence generalizes the notion of statistical convergence of a $$\Delta $$ -measurable real-valued function f defined on a time scale T.

On the Generalized Ideal Asymptotically Statistical Equivalent of Order α for Functions

We introduce new definitions related to the notions of asymptotically Iλ -statistical equivalent of order α to multiple L and strongly Iλ -asymptotically equivalent of order α to multiple L by using two nonnegative real-valued Lebesque measurable functions in the interval (1, ∞) instead of sequences. In addition, we also present some inclusion theorems.

Random Walk in Balls and an Extension of the Banach Integral in Abstract Spaces

We describe the construction of a random walk in a Banach space mathbb {B} with a quasi-orthogonal Schauder basis and show that it is a martingale. Next we prove that under certain additional assumptions the described random walk converges a.s. and in L^{p}left( mathbb {B}right) ,,1le p,<infty , to a random element xi , which generates a probability measure with support contained in the unit ball Bsubset mathbb {B}. Moreover, we define the Banach integral with respect to the distribution of xi for a class of bounded, Borel measurable real-valued functions on B. Next some examples of nonstandard Banach spaces with quasi-orthogonal Schauder bases are presented; furthermore, examples which demonstrate the possibility of applications of all the obtained results in spaces ell ^{p},,1le p < infty and L^{p}[0,1],,1< p < infty are given.

Characterization of intrinsic ultracontractivity for one dimensional Schrödinger operators

We consider a Schrödinger operator L=−d2/dx2+V(x) on R, where V is a real-valued measurable function, and give an explicit and simple characterization of intrinsic ultracontractivity (IU) of the Schrödinger semigroup generated by L for a wide class of potentials. By making use of it, we also give new examples of potentials for which the semigroups satisfy (IU) or non-(IU).

On linearity of pan-integral and pan-integrable functions space

This paper investigates the linearity and integrability of the (+,⋅)-based pan-integrals on subadditive monotone measure spaces. It is shown that all nonnegative pan-integrable functions form a convex cone and the restriction of the pan-integral to the convex cone is a positive homogeneous linear functional. We extend the pan-integral to the general real-valued measurable functions. The generalized pan-integrals are shown to be symmetric and fully homogeneous, and to remain additive for all pan-integrable functions. Thus for a subadditive monotone measure the generalized pan-integral is linear functional defined on the linear space which consists of all pan-integrable functions. We define a p-norm on the linear space consisting of all p-th order pan-integrable functions, and when the monotone measure μ is continuous we obtain a complete normed linear space Lpanp(X,μ) equipped with the p-norm, i.e., an analogue of classical Lebesgue space Lp.

Uncertainty Distribution of Some Composite Uncertain Variables

In this paper, we named the composition by a real-valued measurable function and an uncertain variable as a composite uncertain variable. We focused on the uncertainty distribution for two kinds of composite uncertain variables. The conclusions show: (1) it exists a lower bound when the composed function is continuous and strictly monotonically decreasing at first and then strictly monotonically increasing (e.g. convex downward functions); (2) it exists an upper bound when the composed function is continuous and strictly monotonically increasing at first and then strictly monotonically decreasing (e.g. convex upward functions).

The Landis conjecture for variable coefficient second-order elliptic PDEs

In this work, we study the Landis conjecture for second-order elliptic equations in the plane. Precisely, assume that $V\ge 0$ is a measurable real-valued function satisfying $\|V\|_{L^\infty({\mathbb R}^2)} \le 1$. Let $u$ be a real solution to $\mbox{div}(A \nabla u) - V u = 0$ in ${\mathbb R}^2$. Assume that $|u(z)| \le \exp(c_0 |z|)$ and $u(0) = 1$. Then, for any $R$ sufficiently large, \[ \inf_{|z_0| = R} \|u\|_{L^\infty(B_1(z_0))} \ge \exp(- C R \log R). \] In addition to equations with electric potentials, we also derive similar estimates for equations with magnetic potentials. The proofs rely on transforming the equations to Beltrami systems and Hadamard's three-quasi-circle theorem.

Duality for increasing convex functionals with countably many marginal constraints

In this work we derive a convex dual representation for increasing convex functionals on a space of real-valued Borel measurable functions defined on a countable product of metric spaces. Our main assumption is that the functionals fulfill marginal constraints satisfying a certain tightness condition. In the special case where the marginal constraints are given by expectations or maxima of expectations, we obtain linear and sublinear versions of Kantorovich’s transport duality and the recently discovered martingale transport duality on products of countably many metric spaces.

Definability aspects of the Denjoy integral

Author(s): Walsh, Sean | Abstract: The Denjoy integral is an integral that extends the Lebesgue integral and can integrate any derivative. In this paper, it is shown that the graph of the indefinite Denjoy integral $f\mapsto \int_a^x f$ is a coanalytic non-Borel relation on the product space $M[a,b]\times C[a,b]$, where $M[a,b]$ is the Polish space of real-valued measurable functions on $[a,b]$ and where $C[a,b]$ is the Polish space of real-valued continuous functions on $[a,b]$. Using the same methods, it is also shown that the class of indefinite Denjoy integrals, called $ACG_{\ast}[a,b]$, is a coanalytic but not Borel subclass of the space $C[a,b]$, thus answering a question posed by Dougherty and Kechris. Some basic model theory of the associated spaces of integrable functions is also studied. Here the main result is that, when viewed as an $\mathbb{R}[X]$-module with the indeterminate $X$ being interpreted as the indefinite integral, the space of continuous functions on the interval $[a,b]$ is elementarily equivalent to the Lebesgue-integrable and Denjoy-integrable functions on this interval, and each is stable but not superstable, and that they all have a common decidable theory when viewed as $\mathbb{Q}[X]$-modules.

Comparison meaningful operators and ordinal invariant preferences

The existence of a continuous and order-preserving real-valued function, for the class of continuous and ordinal invariant total preorders, defined on the Banach space of all bounded real-valued functions, which are in turn defined on a given set Ω, is characterized. Whenever the total preorder is nontrivial, the type of representation obtained leads to a functional equation that is closely related to the concept of comparison meaningfulness, and is studied in detail in this setting. In particular, when restricted to the space of bounded and measurable real-valued functions, with respect to some algebra of subsets of Ω, we prove that, if the total preorder is also weakly Paretian, then it can be represented as a Choquet integral with respect to a {0,1}-valued capacity. Some interdisciplinary applications to measurement theory and social choice are also considered.

On the space of (s)-integrable functions for a monotonic measure

In this paper we introduce and study the space of real-valued (s)-integrable functions for a monotonic measure, and consider some relations between this space and the space of real-valued measurable functions.

Vector measures and Mackey topologies

Let Σ be a σ -algebra of subsets of a non-empty set Ω . Let B ( Σ ) be the Banach lattice of all bounded Σ -measurable real-valued functions defined on Ω , equipped with the natural Mackey topology τ ( B ( Σ ) , c a ( Σ ) ) . We study ( τ ( B ( Σ ) , c a ( Σ ) ) , ξ ) -continuous linear operators from B ( Σ ) to a quasicomplete locally convex space ( E , ξ ) . A generalized Nikodym convergence theorem and a Vitali–Hahn–Saks type theorem for operators on B ( Σ ) are obtained. It is shown that the space ( B ( Σ ) , τ ( B ( Σ ) , c a ( Σ ) ) ) has the strict Dunford–Pettis property. Moreover, a Yosida–Hewitt type decomposition for weakly compact operators on B ( Σ ) is given.

A new interpretation of Jensen's inequality and geometric properties of ϕ-means

Abstract We introduce a mean of a real-valued measurable function f on a probability space induced by a strictly monotone function φ. Such a mean is called a φ-mean of f and written by M φ (f). We first give a new interpretation of Jensen's inequality by φ-mean. Next, as an application, we consider some geometric properties of M φ (f), for example, refinement, strictly monotone increasing (continuous) φ-mean path, convexity, etc. Mathematics Subject Classification (2000): Primary 26E60; Secondary 26B25, 26B05.