Projective limits of finite decomposition systems
By U. FEISTE of Greifswald (Eingegangen am 8.12.1975) Summary. Special finite topological decomposition systems were used to get compactifications of topological spaces in [6]. I n this paper the notion of finite decomposition systems is applied for topological measure spaces. We get two canonical topological measure spaces 8 , and X& being projective limits of (discrete) finite decomposition systems for each topological measure space X = (X, Q, A, P) and each net ( & a ) Q E ~ of upward fi1teringfinite:u-algebras in A. 8, is a compact topological measure space and the idea to construct is the same as used in [6]. The compactifications of [a] are cases of some special 8,. Further on we obtain that each measurable set of the remainder of R, has measure zero with respect to the limit measure P, (Theorem 1.). Rd, is the STONE representation space R( IJ Aa) of u A,, hence a BooLean measure spare with regular BOREL measure. Some measure theoretical and topological relations between X, R( u 4.) and R ( 4 ) where R ( 4 ) is the STONE representation space of 4, are given in Theorem 2. and 4. As a corollary from Theorem 2. we get a measure theoretical-topological version to the Theorem of ALEXAN-DROFF HAUSDORFF for compact T, measure spaces R with regular BOREL mcasnre (Theorem 3.).