On modular functions
type: Item
instance of: scholarly article
title: On modular functions
zbMATH Open document ID: 0887.06011
author: Hans Weber
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 1998-05-17T00:00:00Z
review text: For a lattice \(L\) and a group \(G\) a map \(\mu: L\to G\) is called a modular function if for all \(x,y\in L\), we have \(\mu(x\vee y)+ \mu(x\wedge y)= \mu(x)+ \mu(y)\). The important examples that provide much of the motivation for the study of modular functions are furnished by measures on Boolean algebras and linear operators on vector lattices. This paper studies connectedness of the range of a modular function, boundedness of that range when \(G\) is semi-normed, and weak compactness of that range in case \(G\) is a complete locally convex vector space. The paper contains a variety of results with interesting consequences for the motivating examples mentioned above. A typical result is the following (Theorem 4.1 in the present paper). If \(G\) is a complete locally convex linear space and \(\mu: L\to G\) is a modular function then: (i) If \(\mu\) is exhaustive (i.e. \((\mu(a_n))\) is a Cauchy sequence for every increasing sequence \((a_n)\)) then \(\mu(L)\) is relatively weakly compact. (ii) If \(\mu(L)\) is relatively weakly compact and \(\sum_{i\in I}\mu(x_i)- \mu(x_{i- 1})\in \mu(L)- \mu(L)\) for every \(n\in\mathbb{N}\), \(I\subset\{1,\dots, n\}\) and \(x_0< x_1<\cdots< x_n\), then \(\mu\) is exhaustive. Specialized to vector lattices this yields a characterization of o-weakly compact mappings obtained by \textit{P. G. Dodds} [Trans. Am. Math. Soc. 214, 389-402 (1975; Zbl 0289.46010)]. It also contains the well-known result that a Banach space-valued measure is exhaustive if and only if its range is relatively weakly compact. Many proofs in this paper use lattice uniformities, i.e. topologies on the lattice \(L\) that make the lattice operations continuous.
Mathematics Subject Classification ID: 06F30
Mathematics Subject Classification ID: 06F20
Mathematics Subject Classification ID: 28B10
Mathematics Subject Classification ID: 46A40
zbMATH DE Number: 1078850
zbMATH Keywords: exhaustive measure
zbMATH Keywords: modular function
zbMATH Keywords: measures on Boolean algebras
zbMATH Keywords: linear operators on vector lattices
zbMATH Keywords: o-weakly compact mappings
zbMATH Keywords: lattice uniformities
reviewed by: Gerard Buskes
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1078850