Non-additive Measures, Envelopes and Extensions to Quasi-Measures

In the present paper, we introduce the notions of lower envelope and upper envelope for a [0, ∞]-valued function μ defined on a proper sublattice M of a locally complete σ-continuous lattice L, and we extend a finite-stable, supermodular usc-measure μ on a proper sublattice M of L to a quasi*-measure (i.e., a supermodular usc-measure) on $L,$ which is Mδ-inner regular. Analogously, we extend a submodular lsc-measure on M to a quasi*-measure (i.e., a submodular lsc-measure) on $L,$ which is Mσ-outer regular. Furthermore, we have studied notions of measuring envelopes in D-lattices in the context of null-additive, converse null-additive, superadditive and weak converse null-additive functions. 2000 Mathematics Subject Classification. 28A12, 28C15, 28B10