Some properties of algebras of real-valued measurable functions
Let \(\mathcal A\) be a \(\sigma\)-algebra of subsets of a space \(X\). The authors consider the class \({\mathcal M}(X,{\mathcal A})\) of \({\mathcal A}\)-measurable real-valued functions defined on \(X\) as an algebra of functions. (For example, it is well-known that \({\mathcal M}(X,{\mathcal A})\) is a regular ring in the sense of von Neumann.) The authors examine the socle of \({\mathcal M}(X,{\mathcal A})\). They show that, if \(\mathcal A\) contains all singletons, then Soc(\({\mathcal M}(X,{\mathcal A})\)) is equal to the family of all almost zero elements of \({\mathcal M}(X,{\mathcal A})\), and, moreover, it is essential in \({\mathcal M}(X,{\mathcal A})\). They show also that \({\mathcal M}(X,{\mathcal A})\) is an \(\aleph_0\)-self-injective and \(\aleph_0\)-continuous regular ring. In particular, if \(X\) is a Hausdorff space and \(\mathcal B\) is a \(\sigma\)-algebra of Borel sets, then \({\mathcal M}(X,{\mathcal A})\) is \(\aleph_0\)-self-injective and \(\aleph_0\)-continuous regular, and the same properies hold for the algebra \({\mathcal M}(\mathbb{R}^k,{\mathcal L})\), where \({\mathcal L}\) denotes the algebra of Lebesgue measurable subsets of \(\mathbb{R}^k\). Finally, the authors prove that, if \(\mathcal A\) is a \(\sigma\)-algebra that is closed under arbitrary unions, then \({\mathcal M}(X,{\mathcal A})\) is self-injective.
scientific article; zbMATH DE number 5652031
scientific article