ON SETS NOT BELONGING TO ALGEBRAS OF SUBSETS 9
exist three pairwise disjoint sets each of which is not a member of either A\, ^4.2- Moreover,
there do not exist corresponding sets V, Ui,..., Un, J7n+i-
Theore m VI . Consider a countable sequence of a-algebras A\,..., Ak, •., A2 7^ V(X),
such that for every k ^ 2 there exist more than |(fc 1) pairwise disjoint sets not in
Ak, and such that there exist three pairwise disjoint sets each of which is not a member
of either A\, A2- Then there exist pairwise disjoint sets V, Ui,..., Uk, •.. such that if
UkCQ,VDQ = ®, thenQ^Ak.
Theore m VII. (1) Consider a finite sequence of algebras Ai,..., An such that for every
k, 1 k n, there exist more than 4(fc 1) pairwise disjoint sets not members of Ak- Then
there exist pairwise disjoint sets Ui,..., Un, Vi,.. . , Vn such that if Uk C Q, Vk H Q = 0,
then Q ^ *4fc, 1 A T n. (2) The bound 4(fc 1) is best possible in the following sense:
For every natural number n one can construct a sequence of algebras A\,... ,An,An+i
such that for any k, 1 k n, there exist more than 4(fc 1) pairwise disjoint sets not
in Ak, and such that there exist An pairwise disjoint sets not in An-\-i' Moreover, there do
not exist corresponding sets U\,..., Un, Un+\, V\,..., Vn, Vn+\.
Theorem VIII. Consider a countable sequence of a-algebras Ai,..., Ak, •. such that
for every k there exist more than 4(fc 1) pairwise disjoint sets not members of Ak-
Then there exist pairwise disjoint sets U\,..., Uk,..., V\,..., Vk,. such that if Uk C Q,
VkC]Q = 9, thenQ^Ak-
Theore m I X . Consider a finite sequence of algebras Ai,... , An. Suppose there exists a
matrix of pairwise disjoint sets
\u?
...
uzj
where ink §(& 1), Uf Ak- Then there exists pairwise disjoint sets V, U\,..., Un such
that ifUk CQ,V C)Q = 0, then Q £ Ak, 1 k n.
Theorem X . Consider a countable sequence of'o-algebras A\,..., Ak, - •. Suppose there
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