Rendiconti dell'Istituto di Matematica dell'Università di Trieste: an International Journal of Mathematics vol.29 (1998) s.

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https://www.openstarts.units.it/handle/10077/4316

CONTENTS

Y. A. Abramovich, C.D. Aliprantis and O. Burkinshaw
The Invariant Subspace Problem: Some Recent Advances

K. P. S. Bhaskara Rao
Some important theorems in measure theory

W. F. Pfeffer
On Additive Continuous Functions of Figures

Klaus D. Schmidt
Decomposition and Extension of Abstract Measures in Riesz Spaces

Rolf Schneider
Measures in Convex Geometry

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Measures in Convex Geometry
(Università degli Studi di Trieste. Dipartimento di Scienze Matematiche, 1998)
Schneider, Rolf
By convex geometry we understand here the geometry of convex bodies in Euclidean space. In this field, measure theory enters naturally and is useful under several different aspects. First, like in many other fields, measures are employed to quantify the smallness of certain exceptional sets. In our first chapter, we give examples showing how Hausdorff measures of different dimensions are appropriate tools for describing sets of singular points or directions related to the boundary structure of convex bodies. In the second chapter we treat measures that are designed to refflect the local behaviour of convex bodies in a simi¬lar way as curvatures are used in differential geometry. The third connection between convex geometry and measure theory that we want to explain is of an entirely different nature. Here we treat a special class of convex bodies, the zonoids, which can be defined in terms of measures, and we show by an example from stochastic geometry how they are related to other fields. The second of these topics will be treated in greater detail than the other two. Naturally, some facts from the geometry of convex bodies will have to be used without proof. The fundamental notions will be explained and are easy to understand, due to their intuitive character. As a reference where proofs can be found, we mention the book [42].
964 549
Decomposition and Extension of Abstract Measures in Riesz Spaces
(Università degli Studi di Trieste. Dipartimento di Scienze Matematiche, 1998)
Schmidt, Klaus D.
The aim of these notes is to review some recent developments in the theory of abstract measures taking their values in Riesz space. The terni abstract measure is used were to denote a common abstraction of vector measures and linear operators. The topics considered in this survey are: A common approach to vector measures and linear operators, Jordan and Lebesgue decompositions of abstract measures and their applications to vector measures and linear operators, common extensions of linear operators and of vector measures, and extensions of modular functions. We also propose a number of open problems which may stimulate further research in this area. The material of these notes is based on the monograph by Schmidt [5l], two papers by Schmidt and Waldschaks [55], [56], and the PhD Thesis of Waldschaks [60].
795 833
On Additive Continuous Functions of Figures
(Università degli Studi di Trieste. Dipartimento di Scienze Matematiche, 1998)
Pfeffer, W. F.
This is an extended summary of results obtained previously by Z. Buczolich and the author [5]. It describes the relationship between derivatives and variational measures of additive continuous functions of figures, and presents a full descriptive definition of a generalized Riemann integral based on figures.
1067 422
Some important theorems in measure theory
(Università degli Studi di Trieste. Dipartimento di Scienze Matematiche, 1998)
Bhaskara Rao, K. P. S.
In this monograph I shall give several important theorems in measure theory which are not included in any regular graduate/undergraduate courses in measure theory nor are they normally included in standard text books in measure theory. All these theorems are important and have several applications. I shall assume that you know some set theory, some Boolean algebras and some functional analysis. You should definitely know some basic measure theory. Since my aim is to make you familiar with these theorems and their proofs I make no attempt to give the most general versions. Instead, I confine myself to the simplest possible versions without losing the beauty of the proofs.
748 652
The Invariant Subspace Problem: Some Recent Advances
(Università degli Studi di Trieste. Dipartimento di Scienze Matematiche, 1998)
Abramovich, Y. A.
;
Burkinshaw, O.
;
Aliprantis, C. D.
This paper is devoted to recent developments regarding the invariant subspace problem for positive operators on Banach lattices. Some of this material was presented by Y. A. Abramovich at "Workshop di Teoria della Misura e Analisi Reale" Grado (Italia) 18-30 September 1995.
1283 595

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