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On complex Radon measures. II. (English) Zbl 0804.28007
If \(\psi\) is a right continuous complex function of finite variation in \(\mathbb{R}^ n\), then \(\psi\) induces a complex Lebesgue-Stieltjes measure \(\mu_ \psi\) on \(\mathbb{R}^ n\) whose domain is a \(\delta\)-ring \({\mathcal M}_ \psi\) containing all the compact subsets of \(\mathbb{R}^ n\). If \({\mathcal R}\equiv {\mathcal M}_ \psi\cap {\mathcal B}(\mathbb{R}^ n)\), it is well-known that the pair \(({\mathcal M}_ \psi,\mu_ \psi)\) is the Lebesgue completion of the pair \(({\mathcal R},\mu_ \psi|{\mathcal R})\). Conversely, if \(\mu\) is a complex measure on a \(\delta\)-ring \(\mathcal D\) containing the compact subsets of \(\mathbb{R}^ n\), \(\mathcal D\) is the Lebesgue completion of the \(\delta\)-ring \({\mathcal R}\equiv {\mathcal D}\cap {\mathcal B}(\mathbb{R}^ n)\) with respect to the variation \(|\mu|\mid {\mathcal R}\), and \({\mathcal R}= \{E\in {\mathcal B}(\mathbb{R}^ n)\mid |\mu|^*(E)< \infty\}\), then there is a right continuous function \(\psi\) of finite variation on \(\mathbb{R}^ n\) such that \({\mathcal D}= {\mathcal M}_ \psi\) and \(\mu= \mu_ \psi\). This result is essentially the same as Theorem 54.2 of E. J. McShane: “Integration” (1944; Zbl 0060.130), where McShane considers \(\mu\) to be real.
The object of the present paper, which is the continuation of the preceding paper I [the author, Czech. Math. J. 42, No. 4, 599-612 (1992; Zbl 0795.28009)], is to generalize the above-mentioned results to complex Radon measures on a locally compact Hausdorff space \(X\). With each complex continuous linear functional \(\theta\) on the locally convex space \({\mathcal K}(X,\mathbb{C})\) of all continuous complex valued functions on \(X\) with compact support the author associates canonically a unique complex measure \(\mu(\theta)\) defined on a \(\delta\)-ring \({\mathcal M}(\theta)\) containing the \(\delta\)-ring \({\mathcal D}({\mathcal K})\) generated by the family \(\mathcal K\) of all compact subsets of \(X\). The main result of the paper is the following.
Theorem 4.5. Let \(\mathcal D\) be a \(\delta\)-ring containing \({\mathcal D}({\mathcal K})\) and let \(\mu\) be a \({\mathcal D}\)-regular complex measure on \(\mathcal D\). Let \(\nu\equiv\mu|{\mathcal D}({\mathcal K})\) and \(|\nu|\) denote the variation \(v(\nu,{\mathcal D}({\mathcal K}))\). Suppose \({\mathcal R}\equiv {\mathcal B}(X)\cap {\mathcal D}= \{E\in {\mathcal B}(X)\mid |\nu|^ \wedge(E)<\infty\}\), where \(|\nu|^ \wedge\) is the unique Radon- regular extension of \(|\nu|\) to \({\mathcal B}(X)\). If the pair \(({\mathcal D},\mu)\) is the Lebesgue completion of \(({\mathcal R}\), \(\mu({\mathcal R})\) then there exists a unique \(\theta\in {\mathcal K}(X,\mathbb{C})^*\) such that \(\mu= \mu(\theta)\) and \({\mathcal D}= {\mathcal M}(\theta)\). Besides, \(\theta\) is real (respectively, positive) if \(\mu\) is real (respectively, positive).
Making use of the results of the earlier sections the author shows that \({\mathcal K}(X,\mathbb{C})^*\) is isomorphic to the space of all \({\mathcal D}({\mathcal K})\)-regular complex measures on \({\mathcal D}({\mathcal K})\). Finally, he shows that the space of all \(\mathbb{C}\)-valued additive set functions of finite (resp., bounded) variation on a ring of sets is isomorphic to \({\mathcal K}(X,\mathbb{C})^*\) (resp., isometrically isomorphic to \({\mathcal K}(X,\mathbb{C})^*\)) for a suitably chosen totally disconnected locally compact Hausdorff space \(X\).

MSC:
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
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