Ørjan Johansen, Alf B. Rustad
2018 Transactions of the American Mathematical Society  
Quasi-linear functionals are shown to be uniformly continuous and decomposable into a difference of two quasi-integrals. A predual space for the quasi-linear functionals inducing the weak*-topology is given. General constructions of quasi-linear functionals by solid set-functions and q-functions are given. Proposition 15. Let X be a q-space. Let f be a q-function, and let ν be a normalized regular Borel (or topological) measure in X. Define µ on A s by: µC = f (νC); C ∈ C s and µU = f (1) −
more » ... nd µU = f (1) − µ(X\U ); U ∈ O s . If either ν is non-splitting or f is continuous, then µ is a solid set-function. Countable additivity Here we present a generalization to signed topological measures of the proof in [8] that topological measures are countably additive. We recall the following lemmas, stated or implicit in [8], Section 3: Lemma 16 (The Sierpiński Theorem). A compact, connected Hausdorff space cannot be decomposed into a countable family of disjoint, non-empty closed sets. Lemma 17. Let µ be a signed topological measure on X and suppose that C ⊂ X is closed and 0-dimensional. Then the restriction of µ to the closed subsets of C extends to a signed Borel measure on C.
doi:10.1090/s0002-9947-06-03843-8 fatcat:oq73cfrfljgjjbfd6k47wnjhde