Signed Quasi-Measures

Abstract

Abstract. Let X be a compact Hausdorff space and let A denote the subsets of X which are either open or closed. A quasi-linear functional is a map ρ : C(X) → R which is linear on singly generated subalgebras and such that |ρ(f )| ≤ M f for some M < ∞. There is a one-to-one correspondence between the quasi-linear functional on C(X) and the set functions µ :The space of quasi-linear functionals is investigated and quasi-linear maps between two C(X) spaces are studied.Let X be a compact Hausdorff space and C(X) the space of real-valued continuous functions on X. A map ρ : C(X) → R is said to be a quasi-linear functional if ρ is linear on singly generated subalgebras and bounded in the sense that there exists an M < ∞ such that |ρ(f )| ≤ M f u for all f ∈ C(X). Let ρ be the minimal such M . If ρ and η are quasi-linear functionals, we define ρ + η by pointwise action on functions. In this fashion, the collection of all quasi-linear functionals becomes a normed linear space. Call this space QL(X).Notice that if ρ is quasi-linear, and fg = 0, then ρ(f + g) = ρ(f) + ρ(g). In fact, if f and g are also positive, we have that the subalgebra generated by f − g contains both f and g. In general, we can break f and g into positive and negative parts to get the result. Also notice that if c is a constant,Our goal is to find set functions that produce all quasi-linear functionals on C(X). We will use an approach inspired by the techniques in [1] where the theory of positive quasi-linear functionals is presented. We use the notation f ≺ U when U is open to state that 0 ≤ f ≤ 1 and f has support contained in U . We also use the notation sp f for the image of f .Let O be the collection of open sets in X and C the collection of closed sets. Also, let A = O ∪ C. Thus A is the collection of subsets of X which are either open or closed.

Quasi-linear functionals on locally compact spaces

“…Remark 1.19. -If X is compact and µ is finite, condition (TM1) of Definition 1.12 is equivalent, as was noticed in [34], to the following three conditions:…”

“…A. Rustad in [44] first gave a proof of a representation theorem for positive quasi-integrals on functions with compact support when X is locally compact. For a compact space, D. Grubb proved a representation theorem for bounded signed topological measures in [34]. At the time of [34] it was not known whether a signed topological measure is a difference of two topological measures; for such a decomposition on a locally compact space and equivalence of different definitions of signed topological measures see [14].…”

“…This paper, influenced by various works, including [3], [34], [44], combines existing results, improved versions of known results, and new results in a single source for anyone interested in (1) learning about quasi-linear functionals on locally compact noncompact spaces or on compact spaces; (2) further study of quasi-linear functionals, signed quasi-linear functionals, and other related nonlinear functionals; (3) applying quasi-linear functionals in other areas of mathematics.…”

Semisolid sets and topological measures

“…Remark 5.10. If X is compact and µ is finite, condition (T1) of Definition 5.6 is equivalent, as was noticed in [28], to the following three conditions:…”

Дефектные Топологические Меры И Порождаемые Ими Функционалы

“…Они впервые были рассмотрены Д. Дж. Груббом в работе [15], там же показано, что, вообще говоря, |η| не является T -мерой. Таким образом, в представлении η = η 1 − η 2 , где η 1 = (|η|+η)/2, η 2 = (|η|−η)/2, функции η 1 и η 2 не обязаны быть T -мерами.…”