Deficient topological measures and functionals generated by them (Q2848647)

The author studies the quasi-measures defined and studied by \textit{J. F. Aarnes} [Adv. Math. 86, No. 1, 41--67 (1991; Zbl 0744.46052)]. Here \(X\) is a compact Hausdorff space, \(\mathcal{C}, \; \tau\) the classes of closed and open subsets of \(X\), and \(\mathcal{A} = \mathcal{C} \cup \tau\). A mapping \(\psi : \mathcal{A} \to [0, \infty)\) is a DTM (deficient toplogical measure) if it is monotone on \(\mathcal{A}\), additive on \(\mathcal{C}\) and \( \; \tau\), inner regular by compact subsets on \( \tau\) and outer regular by open subsets on \( \mathcal{C}\); \(\psi\) is in M (measure) if it can be extended to a regular Borel measure; \(\psi\) is a TM (toplogical measure) if it is monotone and additive on \(\mathcal{A}\); \(\psi\) is in PDTM if \(\mu \in M, \; \mu \leq \psi\) implies \(\mu = 0.\) In Sections 1, 2, the author proves some properties and inter-relations of these measures. If \(X \subset \mathbb{R}\) then the right and left distribution functions \(\xi_{\psi}^{r}(t)= \psi ([t, \infty) \cap X)\), \(\xi_{\psi}^{l}(t)= \psi (( - \infty, t] \cap X)\) are defined.NEWLINENEWLINENEWLINEFor a general \(X\) and an \(f \in C(X)\), \(f(X)\) is a compact subset of \(\mathbb{R}\) and \(\psi \circ f^{-1}\) is a DTM on \(f(X)\). Using this, and denoting the corresponding \(\xi's\) by \(\xi_{\psi, f}^{r}\), \(\xi_{\psi, f}^{l}\), one gets two Borel measures, uniquely defined by: \(\mu_{\psi, f}^{r}([t, \infty))= \psi(f^{-1} ([t, \infty))\), \(\mu_{\psi, f}^{l}([t, \infty))= \psi(f^{-1} (( - \infty, t])\). In Section 3, using these measures, right and left integrals of \(f\) with respect to \(\psi\) are defined : \(\rho_{\psi}^{r}(f)= \int t d \mu_{\psi, f}^{r}\), \(\rho_{\psi}^{l}(f)= \int t d \mu_{\psi, f}^{l}\). Then the author determines several properties of these functionals \(\rho_{\psi}^{r}\), \(\rho_{\psi}^{l}\) and using them proves some Riesz representation type theorems to retrieve \(\psi\) from them.NEWLINENEWLINESome related additional results and applications are also given.