On additivity of mappings on measurable functions

Abstract

We prove the additivity of regular l-additive mappings T: [InlineMediaObject not available: see fulltext.] → [0,+∞] of a hereditary cone [InlineMediaObject not available: see fulltext.] in the space of measurable functions on a measure space. Some examples are constructed of non-d-additive l-additive mappings T. The monotonicity of l-additive mappings T: [InlineMediaObject not available: see fulltext.] → [0,+∞] is established. The examples are constructed of nonmonotone d-additive mappings T. Let (S, +) be a commutative cancellation semigroup. Given a mapping T: [InlineMediaObject not available: see fulltext.] → S, we prove the equivalence of additivity and l-additivity. It is shown that a strongly regular 2-homogeneous l-subadditive mapping T is subadditive. All results are new even in case [InlineMediaObject not available: see fulltext.] = L∞ +. © 2014 Pleiades Publishing, Ltd.