On integration of vector functions with respect to vector measures

We study integration of Banach space-valued functions with respect to Banach space-valued measures. We focus our attention on natural extensions to this setting of the Birkhoff and McShane integrals. The corresponding generalization of the Birkhoff integral was first considered by Dobrakov under the name S*-integral. Our main result states that S*-integrability implies McShane integrability in contexts in which the later notion is definable. We also show that a function is measurable and McShane integrable if and only if it is Dobrakov integrable (i.e. Bartle *-integrable).     

Integrals of certain random functions with respect to general random measures

For random functions that are sums of random functional series, we determine an integral over a general random measure and prove limit theorems for this integral. We consider the solution of an integral equation with respect to an unknown random measure. National University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 1087–1095, August, 1999.     

Convergence of integrals with respect to L0-valued measures

Translated from Matematicheskie Zametki, Vol. 53, No. 5, pp. 102–106, May, 1993.     

Approximation by polynomials and smooth functions in Sobolev spaces with respect to measures

The density of polynomials is straightforward to prove in Sobolev spaces W^k,p((a,b)), but there exist only partial results in weighted Sobolev spaces; here we improve some of these theorems. The situation is more complicated in infinite intervals, even for weighted L^p spaces; besides, in the present paper we have proved some other results for weighted Sobolev spaces in infinite intervals.     

On the differentiability of Lipschitz functions with respect to measures in the Euclidean space

For every finite measure \({\mu}\) on \({{\mathbb{R}}^n}\) we define a decomposability bundle \({V(\mu,\,\cdot)}\) related to the decompositions of \({\mu}\) in terms of rectifiable one-dimensional measures. We then show that every Lipschitz function on \({{\mathbb{R}}^n}\) is differentiable at \({\mu}\)-a.e. \({x}\) with respect to the subspace \({V(\mu,\,x)}\), and prove that this differentiability result is optimal, in the sense that, following (Alberti et al., Structure of null sets, differentiability of Lipschitz functions, and other problems, 2016), we can construct Lipschitz functions which are not differentiable at \({\mu}\)-a.e. \({x}\) in any direction which is not in \({V(\mu,\,x)}\). As a consequence we obtain a differentiability result for Lipschitz functions with respect to (measures associated to) \({k}\)-dimensional normal currents, which we use to extend certain basic formulas involving normal currents and maps of class \({C^1}\) to Lipschitz maps.     

Real interpolation method on spaces of scalar integrable functions with respect to vector measures

For a given measurable space (Ω,Σ), and a vector measure m:Σ→X with values in a Banach space X we consider the spaces of p-power integrable and weakly integrable, respectively, functions with respect to the measure m, Lp(m) and , for 1?p<∞. In this note we describe the real interpolated spaces that we obtain when the K-method is applied to any couple of these spaces.     

Singularity of self-similar measures with respect to Hausdorff measures

Besicovitch (1934) and Eggleston (1949) analyzed subsets of points of the unit interval with given frequencies in the figures of their base- expansions. We extend this analysis to self-similar sets, by replacing the frequencies of figures with the frequencies of the generating similitudes. We focus on the interplay among such sets, self-similar measures, and Hausdorff measures. We give a fine-tuned classification of the Hausdorff measures according to the singularity of the self-similar measures with respect to those measures. We show that the self-similar measures are concentrated on sets whose frequencies of similitudes obey the Law of the Iterated Logarithm.

     

Elliptic operators with unbounded diffusion coefficients in L 2 spaces with respect to invariant measures

We study the self-adjoint and dissipative realization A of a second order elliptic differential operator with unbounded regular coefficients in , where μ(dx) = ρ (x)dx is the associated invariant measure. We prove a maximal regularity result under suitable assumptions, that generalize the well known conditions in the case of constant diffusion part. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Differentiability of integrals of real functions with respect toL 0-valued measures

We obtain conditions for the convergence of expressions inL ₀ as the setA decreases. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1582–1585, November, 1999.     

The topological asymptotic with respect to a singular boundary perturbation

The aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the insertion of a small hole in the domain. The question that we address here is what happens if the hole is located at the boundary of the domain and what happens if the boundary is not regular. The adjoint method and the domain truncation technique are proposed to solve this problem. As a model example, we consider the Laplace equation in a domain with a corner. To cite this article: B. Samet, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Polynomial expansions of Boolean functions with respect to nondegenerate functions

Translated from Algebra i Logika, Vol. 30, No. 6, pp. 631–637, November–December, 1991.     

Norming sets and integration with respect to vector measures

Let v be a countably additive measure defined on a measurable space (Ω, Σ) and taking values in a Banach space X. Let f : Ω → ? be a measurable function. In order to check the integrability (respectively, weak integrability) of f with respect to v it is sometimes enough to test on a norming set Λ ⊂ X^*. In this paper we show that this is the case when A is a James boundary for BX^* (respectively, Λ is weak^*-thick). Some examples and applications are given as well.     

Dentability indices with respect to measures of non-compactness

We study the relationships between the ordinal indices of set derivations associated to several measures of non-compactness. We obtain applications to the Szlenk index, improving a result of Lancien, and LUR renorming, providing a non-probabilistic proof of a result of Troyanski.     

An integral with respect to probabilistic-valued decomposable measures

Several concepts of approximate reasoning in uncertainty processing are linked to the processing of distribution functions. In this paper we make use of probabilistic framework of approximate reasoning by proposing a Lebesgue-type approach to integration of non-negative real-valued functions with respect to probabilistic-valued decomposable (sub)measures. Basic properties of the corresponding probabilistic integral are investigated in detail. It is shown that certain properties, among them linearity and additivity, depend on the properties of the underlying triangle function providing (sub)additivity condition of the considered (sub)measure. It is demonstrated that the introduced integral brings a new tool in approximate reasoning and uncertainty processing with possible applications in several areas.     

The Weighted L 1-integrability of functions and the parseval equality with respect to multiplicative systems

In this paper we prove necessary and sufficient conditions for the weighted L ¹-integrability of functions defined on [0, 1) in terms of Fourier coefficients with respect to a multiplicative system of bounded type. These results are counterparts of trigonometric ones obtained by M. and S. Izumi and M. M. Robertson.     

Smoothness of functions with respect to a set

In this note we consider the properties of functions which are smooth with respect to congruent sets. We find necessary and sufficient conditions which these congruent sets must satisfy in order that asymptotic differentiability, together with the smoothness relative to a set, will imply ordinary differentiability almost everywhere. We consider the case of measurable and continuous functions.Translated from Matematicheskie Zametki, Vol. 15, No. 3, pp. 431–436, March, 1974.