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.