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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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##### References:
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