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Deficient topological measures and functionals generated by them. (English. Russian original) Zbl 1279.28018
Sb. Math. 204, No. 5, 726-761 (2013); translation from Mat. Sb. 204, No. 5, 109-142 (2013).
The author studies the quasi-measures defined and studied by 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.
For 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.
Some related additional results and applications are also given.

##### MSC:
 28C15 Set functions and measures on topological spaces (regularity of measures, etc.) 28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
Zbl 0744.46052
Full Text:
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