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Control and separating points of modular functions. (English) Zbl 0965.28004
Let $$L$$ be a lattice, $$X$$ a locally convex linear space and $$\mu \:L\to X$$ a modular function, i.e., a function satisfying $\mu (x\vee y)+\mu (x\wedge y)=\mu (x)+\mu (y) .$ Let $$u(\mu)$$ denote the $$\mu$$-uniformity, i.e., the weakest uniformity on $$L$$ which makes the lattice operations and $$\mu$$ uniformly continuous. In the first part of the paper the authors study the problems, (i) when for a modular function $$\mu \:L\to X$$ there is a real valued modular function $$\nu$$ on $$L$$ with $$u(\nu)=u(\mu)$$ and (ii) when for a set $$M$$ of $$X$$-valued modular functions there is a modular function $$\nu \: L\to X$$ with $$u(\nu)$$=sup$$\{u(\mu)\:\mu \in M\}$$. Such a function $$\nu$$ is called a control for $$\mu$$ or for $$M$$, respectively. The authors present several results concerning controls on complemented lattices generalizing known results about control measures on Boolean algebras. In the second part of the paper the authors study the problem when a sequence $$\mu_n$$ of group-valued modular functions on a complemented lattice $$L$$ has a separating point, i.e., a point $$a\in L$$ such that $$\mu_n(a)\neq \mu_m(a)$$ for $$n\neq m$$. The presented results were obtained by A. Basile and H. Weber [Rad. Mat. 2, 113-125 (1986; Zbl 0596.28015)] in the case of $$L$$ being a Boolean algebra.
Reviewer: Hans Weber (Udine)

##### MSC:
 28B05 Vector-valued set functions, measures and integrals 06B30 Topological lattices 06C15 Complemented lattices, orthocomplemented lattices and posets 28B10 Group- or semigroup-valued set functions, measures and integrals
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