We consider the general setting of A.D. Alexandroff, namely, an arbitrary setXand an arbitrary lattice of subsets ofX,?.??(?)denotes the algebra of subsets ofXgenerated by?andMR(?)the set of all lattice regular, (finitely additive) measures on??(?).First, we investigate various topologies onMR(?)and on various important subsets ofMR(?), compare those topologies, and consider questions of measure repleteness whenever it is appropriate.Then, we consider the weak topology onMR(?), mainly when?isδand normal, which is the usual Alexandroff framework. This more general setting enables us to extend various results related to the special case of Tychonoff spaces, lattices of zero sets, and Baire measures, and to develop a systematic procedure for obtaining various topological measure theory results on specific subsets ofMR(?)in the weak topology with?a particular topological lattice.
展开▼
