4 L.S. GRINBLAT
Claim 1.1. Let Ai,... , An be a finite sequence of algebras such that for every k, 1 k
n, there exist at most |(fc 1) pairwise disjoint sets which are not members of Ak- Then
there exists a set which is not a member of any Ah, I k n.
Claim 1.2. Let A\,..., Ah, be a countable sequence of o-algebras such that for every
k there exist at most |(& 1) pairwise disjoint sets which are not members of Ak. Then
there exists a set which is not a member of any Ak-
Let us return to the algebras A[, A'2, A'3 constructed above. It follows from Claim 1.1
(and also from Claim 1.2) that if there existed not two but at least three pairwise disjoint
sets not belonging to ^.3, then there would exist a set not a member of A[, A'2, A'3. This is
no accident: We shall show that the bound |(fc 1) is in a certain sense the best possible.
Claim 1.2 is a generalization of the following theorem of Gitik and Shelah [GS]:
Theorem . Let /J,\ , . . . , //&,... be a countable sequence of two-valued measures defined on
a set of the power of the continuum. Then there exists a set which is fj,k~nonmeasurable
for all k.5
iFvom the standpoint of the technique proposed in this memoir, the difficulty of Gitik
and Shelah's theorem is not that one can construct a model in which there exists a two-
valued measure //, defined on a set of the power of the continuum, such that there do
not exist ^1 pairwise disjoint /x-nonmeasurable sets. The difficulty is as follows. One can
construct a model (see [BD]) in which there exists a countable family of ultrafiliters {a\}
over a set of the power of the continuum with the following property: define a set function
fi by stipulating that fJ(M) = 0 if and only if M £ a\ for all //, and /i(M) = 1 if and only
if M G a\ for all A; then /1 is a two-valued measure. Gitik and Shelah themselves used
forcing to prove their theorem. A purely combinatorial proof was proposed by Fremlin and
in [K]. We shall use the following equivalent version of the Gitik-Shelah theorem:
Theorem. Let JJL\ , . . . , ^ , . . . be a countable sequence of two-valued measures defined on
a set of the power of the continuum. Then there exist pairwise disjoint sets Mi,..., M&,...
such that Mk is fik-nonmeasurable.
5 This theorem is clearly a generalization of the Alaoglu-Erdos theorem - undoubtedly a profound and
far from trivial generalization.
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