Get A Source Book in Mathematics: volume 1 PDF

By David Eugene Smith

ISBN-10: 0486605523

ISBN-13: 9780486605524

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Let T be the set of all tk. We know that ~H n T = 0. S. G R I N B L A T For each z £ Z*, consider a set Uz G z. For each set Q\ of the third kind, consider a set Qk,i C Q\; Qk,i $. Ak- By the same reasoning as in Section 5, we may assume that the sets Uz, Qk,i constitute a family of pairwise disjoint sets. 2, we may assume that for any set Uz and any algebra Ak, either Uz i Ah or V(UZ) C A - 2 7 We can already define some of the required sets. If s\ = z G Z*, then 0? = uz. Step II. Let tk G T and tjt 9 (J m ,i- We may assume that this is possible only when *k G Q m ) i n k e r ^ m and Xm>i = Qm i ^ ker*4 m is an outer separable set, all points of which are regular.

Hence there exists Un G hn such that C A. Thus all elements of V([Jn Un) are members of A. But this contradicts the fact that ( J Un D ~H D (U 0 ker A) + 0. 2. , [G]). ON SETS NOT BELONGING TO ALGEBRAS OF SUBSETS 39 An algebra A is unsaturated if and only if |ker^4| < N0- If A = V(X), then A is unsaturated and ker*4 = 0. 4|. 3. A simple algebra A which is not unsaturated will be called a strictly simple algebra. The following assertion is obvious: A s s e r t i o n . An almost a-algebra A is strictly simple if and only if there exist No pairwise disjoint sets Mi,...

Let Z' — {z G Z | z is an irregular point}, Z" = Z\Z'. It is obvious that all points Z" are regular, Z 77 D k e r ^ Z 7 ^ 0, Z" C PX\X, and for every point z G Z" one can construct a function / , continuous on /3X, such that f(z) and f(x) = 0 1 > 0 for x G X. Let U C X, Z HU = 0, U n ker A ± 0, zv = z n TJ = z" n u. Since A is an almost cr-algebra, consider a two-valued measure \sPA defined on X by putting \ivA{M) = 0 iff V(M H U) C A. Obviously, fi%(M) = 0 if and only if M fl U fl ker A = 0. Now consider a countable sequence of strictly separable almost cr-algebras A\,...

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A Source Book in Mathematics: volume 1 by David Eugene Smith


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