Projective limits of finite decomposition systems

Special finite topological decomposition systems were used to get compactifications of topological spaces in [6]. In this paper the notion of finite decomposition systems is applied for topological measure spaces. We get two canonical topological measure spaces X ∞ and X ∞ d being projective limits of (discrete) finite decomposition systems for each topological measure space X = ( X, O, A, P ) and each net ( A α ) α ϵ I of upward filtering finite σ‐algebras in A . X ∞ is a compact topological measure space and the idea to construct is the same as used in [6]. The compactifications of [6] are cases of some special X ∞ . Further on we obtain that each measurable set of the remainder of X ∞ has measure zero with respect to the limit measure P ∞ (Theorem 1). X ∞ d is the STONE representation space X ( \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document} ) of \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document} A α , hence a Boolean measure space with regular Borel measure. Some measure theoretical and topological relations between X , X ( \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document} ) and x(A) where x(A) is the Stone representation space of A , are given in Theorem 2. and 4. As a corollary from Theorem 2. we get a measure theoretical‐topological version to the Theorem of Alexandroff Hausdorff for compact T 2 measure spaces x with regular Borel measure (Theorem 3.).