Unbounded weighted Radon measures and dual of certain function spaces with strict topology (Q1941004)

\textit{R. C. Buck} introduced in [Mich. Math. J. 5, 95--104 (1958; Zbl 0087.31502)] the strict topology on the space \(C_0(X)\) of all real-valued functions on a locally compact space \(X\) vanishing at infinity, and represented its dual as a space of measures. For generalizations see, e.g., [\textit{R. F. Wheeler}, Expo. Math. 1, 97--190 (1983; Zbl 0522.28009)]). In the present paper, the authors introduce strict topologies on the spaces \(C_b(X,\omega)\) and \(\tilde{C}_b(X,\omega)\) and determine their duals. Here, \(X\) is a Hausdorff space such that the space \(C_b(X)\) of bounded real-valued continuous functions separates the points of \(X\), \(\omega:X\rightarrow (0,+\infty[\) is Borel measurable such that \(1/\omega\) is bounded on the compact subsets of \(X\), \(C_b(X,\omega)\) is the space of functions \(f:X\rightarrow\mathbb{R}\) with \(f/\omega\in C_b(X)\), and \(\tilde{C}_b(X,\omega)\) is the space of continuous functions \(f:X\rightarrow\mathbb{R}\) such that \(f/\omega\) is bounded.