Radon, Baire, and Borel measures on compact spaces. II (Q923315)

In Part I of this paper [Hokkaido Math. J. 18, 231-249 (1989; Zbl 0696.46025)], the author begins with the space C(X) of continuous functions on a compact space X and constructs from scratch the spaces \(B(X)\) and \(\tilde B(X)\) of bounded Borel and bounded Baire functions, respectively, on X. It was then shown how these spaces can be imbedded as Banach sublattices of the second dual space \(C''(X)\) of \(C(X)\) without the aid of the Riesz representation theorem. The author's intent is to render as transparent as possible the relations between the spaces of continuous, Baire and Borel functions and of Radon, Baire and Borel measures. Purely functional analytic means are employed in this task. In Part II, under review here, the author focuses on measures. A Radon measure is defined to be a norm bounded linear form on \(C(X)\), a Borel measure is an order \(\sigma\)-continuous linear form on \(B(X)\), and, completely analogously, a Baire measure is defined to be an order \(\sigma\)-continuous linear form on \(\tilde B(X)\) (that this is equivalent to the usual notion of a Baire measure is discussed). One main result (Theorem 2.3) is that a linear form on B(X) is a regular Borel measure if and only if it is continuous in the \({\mathcal I}\) topology (of uniform convergence on the order intervals of the dual space of C(X)) on \(C''(X)\). Several corollaries, most important of which is the Riesz- Markov representation theorem (that every Radon measure on X has a unique regular Borel extension) are given. In addition, it is proved that a Borel measure is regular if and only if it is \({\mathcal I}\)-continuous on the space \(B_ 0(X)\). \((B_ 0\) is the first step in the construction of \(B(X)\).) The relation between Baire measures and \({\mathcal I}\)-continuity as well as the regularity of Baire measures (they all are) are discussed. In the final section, the inclusions \(C(X)\subseteq \tilde B(X)\subseteq B(X)\subseteq C''(X)\) are considered and conditions are given under which some or all of the inclusions are not proper. For example, \(C(X)=\tilde B(X)\) if and only if X is finite and \(\tilde B(X)=B(X)\) if and only if every open subset of X is an \(F_{\sigma}\) set.