Topological measure theory: A study of repleteness and measure repleteness.

摘要

Let {dollar}{lcub}cal L{rcub}{dollar} be a lattice of subsets of the abstract set {dollar}X{dollar}, and {dollar}{lcub}cal A{rcub}{dollar}({dollar}{lcub}cal L{rcub}{dollar}), the algebra generated by {dollar}{lcub}cal L{rcub}{dollar}. We define {dollar}M{dollar}({dollar}{lcub}cal L{rcub}{dollar}) to be all non-negative, finitely additive, finite valued measures on {dollar}{lcub}cal A{rcub}{dollar}({dollar}{lcub}cal L{rcub}{dollar}; {dollar}Msbsigma{dollar}({dollar}{lcub}cal L{rcub}{dollar}) the measures of {dollar}M{dollar}({dollar}{lcub}cal L{rcub}{dollar}) which are {dollar}sigma{dollar}-smooth on {dollar}{lcub}cal L{rcub}{dollar} and {dollar}Msb{lcub}R{rcub}{dollar}({dollar}{lcub}cal L{rcub}{dollar}) those measure of {dollar}M{dollar}({dollar}{lcub}cal L{rcub}{dollar}) which are {dollar}{lcub}cal L{rcub}{dollar}-regular. If {dollar}mu{dollar} {dollar}in{dollar} {dollar}Msb{lcub}R{rcub}{dollar}({dollar}{lcub}cal L{rcub}{dollar}) and {dollar}mu{dollar} {dollar}in{dollar} {dollar}Msbsigma{dollar}({dollar}{lcub}cal L{rcub}{dollar}), we write {dollar}mu{dollar} {dollar}in{dollar} {dollar}Msbsp{lcub}R{rcub}{lcub}sigma{rcub}{dollar}({dollar}{lcub}cal L{rcub}{dollar}) to represent the countably additive {dollar}{lcub}cal L{rcub}{dollar}-regular measures of {dollar}M{dollar}({dollar}{lcub}cal L{rcub}{dollar}). {dollar}Msbtau{dollar}({dollar}{lcub}cal L{rcub}{dollar}) designates those {dollar}mu{dollar} {dollar}in{dollar} {dollar}M{dollar}({dollar}{lcub}cal L{rcub}{dollar}) which are {dollar}tau{dollar}-smooth on {dollar}{lcub}cal L{rcub}{dollar}. If in any of the above definitions, we restrict the measures to be only zero-one valued, the corresponding set is denoted by an {dollar}I{dollar}. Thus {dollar}Isb{lcub}R{rcub}{dollar}({dollar}{lcub}cal L{rcub}{dollar}) are the 0-1 valued measures of {dollar}Msb{lcub}R{rcub}{dollar}({dollar}{lcub}cal L{rcub}{dollar}), and {dollar}Isbsp{lcub}R{rcub}{lcub}sigma{rcub}{dollar}({dollar}{lcub}cal L{rcub}{dollar}) are the 0-1 valued measures of {dollar}Msbsp{lcub}R{rcub}{lcub}sigma{rcub}{dollar}({dollar}{lcub}cal L{rcub}{dollar}) and so on.; In the first part of this dissertation, we investigate various topological properties associated with a lattice {dollar}{lcub}cal L{rcub}{dollar}, such as {dollar}{lcub}cal L{rcub}{dollar} normal, regular, etc. ..., and express these properties in terms of measures from the {dollar}I{dollar}'s. We also do this for pairs of lattices {dollar}{lcub}cal L{rcub}sb2{dollar} {dollar}subset{dollar} {dollar}{lcub}cal L{rcub}sb1{dollar} thereby extending work of (12), (5), (7) and (2). These results are then systematically applied to the Wallman Space {dollar}Isb{lcub}R{rcub}{dollar}({dollar}{lcub}cal L{rcub}{dollar}) and the lattices {dollar}W{dollar}({dollar}{lcub}cal L{rcub}{dollar}) and {dollar}tau W{dollar}({dollar}{lcub}cal L{rcub}{dollar}). In particular, conditions for {dollar}tau W{dollar}({dollar}{lcub}cal L{rcub}{dollar}) to be normal are thoroughly investigated, a similar investigation is carried our for the space {dollar}Isbsp{lcub}R{rcub}{lcub}sigma{rcub}{dollar}({dollar}{lcub}cal L{rcub}{dollar}) with the relative Wallman topology.; In addition, in the first part of the dissertation we give conditions in {dollar}{lcub}cal L{rcub}{dollar} under which {dollar}mu{dollar} {dollar}in{dollar} {dollar}Msbsigma{dollar}({dollar}{lcub}cal L{rcub}{dollar}) or {dollar}mu{dollar} {dollar}in{dollar} {dollar}Msbsigma{dollar}({dollar}{lcub}cal L{rcub}spprime{dollar}) or {dollar}mu{dollar} {dollar}in{dollar} {dollar}Msbtau{dollar}({dollar}{lcub}cal L{rcub}spprime{dollar}) (where {dollar}{lcub}cal L{rcub}spprime{dollar} is the complement lattice) are in {dollar}Msb{lcub}R{rcub}{dollar}({dollar}{lcub}cal L{rcub}{dollar}).; In the second part of the dissertation, the remainders {dollar}Isb{lcub}R{rcub}{dollar}({dollar}{lcub}cal L{rcub}{dollar}) {dollar}-{dollar} {dollar}X{dollar},