By J. R. Choksi, R. R. Simha (auth.), Alexandra Bellow, Dietrich Kölzow (eds.)
Read Online or Download Measure Theory: Proceedings of the Conference Held at Oberwolfach, 15–21 June, 1975 PDF
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Extra info for Measure Theory: Proceedings of the Conference Held at Oberwolfach, 15–21 June, 1975
Example text
Assume that v = ~I Z is compact and let = Z~ ~ be a countably multiplicative compact class approximating Z~. then that from ~(r) with respect to v . ) a countable algebra contained in Z ~ such E c ~(E) and all a(~(E))-atoms outside fo some v-null set a(~E)) are elements of ~ (in view of Lemma, such an algebra always exists). Then take an arbitrary net of countable algebras ~ y r ~ , y E P such that ~ o c~,i c ... ~ y c ... = where 9 and ~y a(~<~ 1~_Y) , put (U , u ~ E P . Then take a countable algebras Z ~ ~y = ~= , ~y c ~y c y and such that c 2 c ...
We h a v e the theorem: For such ( X , ~ ) , e~uivalent statements all hold. and g the above For, in (5) above, the maps must be of bounded Borel class, by a recent result of f 45 D. Preiss [6]; hence, by a theorem of R. Hansell limits of polntwise classes, convergent f and g invariant on locally compact groups! null set had to be dealt with. , that the "bad" set each "vertical" ~x] x I isomorphism of T that his argument T onto I I commutes (where standard set, Unfortunately pointed Borel. exists say a Borel I x I, existence However, @ P r if there exists a to verticals -- that is, the diagram I This would follow at once if (for isomorphism such that @ of @ preserves B onto a verticals.
We now mention two qestions for which we do not know the answers: QI. It the restriction of a compact meaure to any sub-u-algebra compact? (The answer to this questioois unknown even in the case of Lebesgue measure on Borel s e ~ o f the unit interval). Q2. Let ( X , ~ ) and ( Y , ~ ) be measurable spaces and let be a compact measure on the G-algebra o(~x~ ) . Are the marginals of ~ compact ? 42 References. G. , On the spaces with perfect but non-compact measures (in Russian), Naucn. map. Taskent, Institut Nat.