MR3191427 Naralenkov, Kirill M., A Lusin type measurability property for vector- valued functions. J. Math. Anal. Appl. 417 (2014), no. 1, 293307. 28A20

In the paper under review the author introduces the notion of Riemann measurability for vector-valued functions, generalizing the classical Lusin condition, which is equivalent to the Lebesgue measurability for real valued functions. Let X be a Banach space, let f : [a; b] ! X and let E be a measurable subset of [a; b]. The function f is said to be Riemann measurable on E if for each " > 0 there exist a closed set F E with (E n F) < 0 (where is the Lebesgue measure) and a positive number such that k XK k=1 ff(tk) ?? f(t0 k)g (Ik)k < " whenever fIkgKk =1 is a nite collection of pairwise non-overlapping intervals with max1 k K (Ik) < and tk; t0 k 2 Ik T F. The Riemann measurability is more relevant to Riemann type integration theory, such as those of McShane and Henstock, rather than the classical notion of Bochner or scalar measurability. In par- ticular the author studies the relationship between the Riemann measurability and the M and the H integrals that are obtained if we assume that the gauge in the de nitions of McShane and Henstock integral can be chosen Lebesgue measurable. The main results are the following If f : [a; b] ! X is H-integrable on a measurable subset E of [a; b], then f is Riemann measurable on E. If f : [a; b] ! X is both bounded and Riemann measurable on a measurable subset E of [a; b], then f is M-integrable on E. If f : [a; b] ! X is both Riemann measurable and McShane (Henstock) integrable on a measurable subset E of [a; b], then f is M-integrable (H-integrable) on E. Suppose X separable. If f : [a; b] ! X is McShane (Henstock) integrable, then f is M-integrable (H-integrable.) The author concludes the paper with the following open problem: for which families of non-separable Banach spaces does the McShane (or even the Pettis) integrability imply Riemann measurability? Reviewed by (L. Di Piazza

Variational measures in the theory of integration

{Variational measures in the theory of integration} {Luisa Di Piazza} {Palermo , Italy} We will present here some results concerning the variational measures associated to a real valued function, or, in a more general setting, to a vector valued function. Roughly speaking, given a function $\Phi$ defined on an interval $[a,b]$ of the real line it is possible to construct, using suitable families of intervals, a measure $\mu_{\Phi}$ which carries information about $\Phi$. If $\Phi$ is a real valued function, then the $\sigma$-finiteness of the measure $\mu_{\Phi}$ implies the a.e. differentiability of $\Phi$, while the absolute continuity of the measure $\mu_{\Phi}$ characterizes the functions $\Phi$ which are Henstock-Kurzweil primitives. The situation becomes more complicate if we consider functions taking values in an infinite dimensional Banach space. If the Banach space has the Radon-Nikod\'{y}m property, then it is possible to obtain properties similar to those of the real case. But it is surprising that by means of the variational measures it is possible to characterize the Banach space having the Radon-Nikod\'{y}m property. \begin{thebibliography}{99} \bibitem{bds1} B. Bongiorno, L. Di Piazza and V. Skvortsov, \textit{ A new full descriptive characterization of Denjoy-Perron integral}, Real Analysis Exchange, {\bf 21} (1995/96), 256--263. \bibitem{bdm} B. Bongiorno, L. Di Piazza and K. Musial, \textit{ A characterization of the Radon-Nikod\'{y}m property by finitely additive interval functions}, Illinois Journal of Mathematics. Volume 53, Number 1 (2009), 87-99. \bibitem{db} D. Bongiorno, \textit{ Stepanoff's theorem in separable Banach spaces}, Comment. Math. Univ. Ca\-ro\-linae, {\bf 39} (1998), 323--335. \bibitem{ldp1} L. Di Piazza, \textit{ Varational measures in the theory of the integration in $R^m$}, Czechos. Math. Jour. 51(126) (2001), no. 1, 95--110. \bibitem{vm} V. Marraffa, \textit{ A descriptive characterization of the variational Henstock integral}, Proceedings of the International Mathematics Conference (Manila, 1998), Matimy\'{a}s Mat. {\bf 22} (1999), no. 2, 73--84

MR2817222 Ursescu, Corneliu, A mean value inequality for multifunctions. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 54(102) (2011), no. 2, 193–200

The paper is devoted to extend some mean value inequalities from the function setting to the multifunction one. Let (M,d) be a metric space, let F be a multifunctions defined on D \subset R and taking values in the family of nonempty subsets of M, and let g: D\rightarrow R be a strictly increasing function. The author proves the following inequality: \frac{\delta(F(b),F(a))}{g(b)-g(a)} \leq \sup_{s\in [a,b)\cap D} \sup_{S\in F(s)} \sup_{t\in (s,b)\cap D} \frac{\delta(F(t),S)}{g(t)-g(s)}, where a and b are two points of D with a<b and, if Q and P are nonempty subsets of M, then \delta(Q,P)=\sup_{p\in P} \inf_{q\in Q}d(q,p). An application of the previous inequality to the Dini derivatives of a multifunction is also given. Reviewed by L. Di Piazz

Set valued Kurzweil-Henstock-Pettis integral

It is shown that the obvious generalization of the Pettis integral of a multifunction obtained by replacing the Lebesgue integrability of the support functions by the Kurzweil--Henstock integrability, produces an integral which can be described -- in case of multifunctions with (weakly) compact convex values -- in terms of the Pettis set-valued integral

Integrals and Banach spaces for finite order distributions

summary:Let $\mathcal B_c$ denote the real-valued functions continuous on the extended real line and vanishing at $-\infty$. Let $\mathcal B_r$ denote the functions that are left continuous, have a right limit at each point and vanish at $-\infty$. Define $\mathcal A^n_c$ to be the space of tempered distributions that are the $n$th distributional derivative of a unique function in $\mathcal B_c$. Similarly with $\mathcal A^n_r$ from $\mathcal B_r$. A type of integral is defined on distributions in $\mathcal A^n_c$ and $\mathcal A^n_r$. The multipliers are iterated integrals of functions of bounded variation. For each $n\in \mathbb N$, the spaces $\mathcal A^n_c$ and $\mathcal A^n_r$ are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to $\mathcal B_c$ and $\mathcal B_r$, respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable. The dual space is isometrically isomorphic to the functions of bounded variation. The space $\mathcal A_c^1$ is the completion of the $L^1$ functions in the Alexiewicz norm. The space $\mathcal A_r^1$ contains all finite signed Borel measures. Many of the usual properties of integrals hold: Hölder inequality, second mean value theorem, continuity in norm, linear change of variables, a convergence theorem

Representations of multimeasures via multivalued Bartle-Dunford-Schwartz integral

An integral for a scalar function with respect to a multimeasure $N$ taking its values in a locally convex space is introduced. The definition is independent of the selections of $N$ and is related to a functional version of the Bartle-Dunford-Schwartz integral with respect to a vector measure presented by Lewis. Its properties are studied together with its application to Radon-Nikodym theorems in order to represent as an integrable derivative the ratio of two general multimeasures or two $d_H$-multimeasures; equivalent conditions are provided in both cases.Comment: we have changed the title of the articl

Convergence for varying measures in the topological case

In this paper convergence theorems for sequences of scalar, vector and multivalued Pettis integrable functions on a topological measure space are proved for varying measures vaguely convergent.Comment: 19 page

Linear Dynamics Induced by Odometers

Weighted shifts are an important concrete class of operators in linear dynamics. In particular, they are an essential tool in distinguishing variety dynamical properties. Recently, a systematic study of dynamical properties of composition operators on $L^p$ spaces has been initiated. This class of operators includes weighted shifts and also allows flexibility in construction of other concrete examples. In this article, we study one such concrete class of operators, namely composition operators induced by measures on odometers. In particular, we study measures on odometers which induce mixing and transitive linear operators on $L^p$ spaces.Comment: 15 pages, keywords: linear dynamics, composition operators, topological mixing, topological transitivity, odometer