On modular functions.

*(English)*Zbl 0887.06011For 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 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.

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 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.

Reviewer: G.Buskes (University)

##### MSC:

06F30 | Ordered topological structures (aspects of ordered structures) |

06F20 | Ordered abelian groups, Riesz groups, ordered linear spaces |

28B10 | Group- or semigroup-valued set functions, measures and integrals |

46A40 | Ordered topological linear spaces, vector lattices |