ON SETS NOT BELONGING TO ALGEBRAS OF SUBSETS 7
can construct a sequence of algebras A\,... , An, An+i such that if k n, there exist more
than | ( k 1) pairwise disjoint sets not in Ah, and such that there are only [^p] 8 pairwise
disjoint sets not in An-\-\. Moreover, there does not exist a set which does not belong to
all the algebras Ah, \ k n -\- 1.
Incidentally, our construction of the algebras A[, A'2, A'3 in the Introduction actually
yields a proof of part (2) of Theorem I in the case n = 2.
Before stating Theorem II, which is a generalization of Claim 1.2, we need the following
Definition 2.1 An algebra A is called an almost cr-algebra if for any countable sequence
M i , . . . , Mjt,... C X, such that for any fc, V(Mk) C A, it is also true that A 3 (Jk Mk.
Theore m II. Consider a countable sequence of almost a-algebras Ai,... , Ak, , A2 7^
V{X), such that for every k / 2 there exist more than |(fc 1) pairwise disjoint sets not
in Ak- Then there exists Q ^ Ak for all k.
As will be shown below (Example 6.1), Theorem II is no longer true if we do not demand
that the algebras Ak be almost a-algebras.
Theore m III. (1) Consider an at most countable sequence of a-algebras Ai,... , Ak, •.
such that there exists a matrix of pairwise disjoint sets9
\ /
(each row from the third row on contains two sets) for which Uf £ Ak- Then there exists
a set U £ Ak for all k. (2) If not two but three rows of the matrix contain one set each,
then the set U need not exist.
Part (2) of Theorem III was proved in the Introduction. The algebras A\, A'2- A'3
constructed there are cr-algebras. There exist pairwise disjoint sets Ui, U2, U3 such that
Ui A'i, and there exists no set which is not a member of any of these algebras.
8
[a] denotes the largest integer a.
9
That is, a matrix of sets such that any two different sets in the matrix are disjoint.
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