The "proof of the equivalent of the Gitik-Shelah theorem and the last statement is very
easy. Indeed, let / / i , . . . , ^ , . . . be a countable sequence of two-valued measures defined
on a set X of the cardinality of the continuum. By the Gitik-Shelah theorem, there exists
a set M\ which is //fc-nonmeasurable for all k. On the set X\M\ we define two-valued
measures / i j , . . . , ^ , . . . as follows: jilk(M) = 1 if and only if there exists a set M ' such
that fik(M') = 1 and M = (X\Mi) 0 M'. 6 By the Gitik-Shelah theorem, there exists a
set M
C X\M\ which is ^-nonmeasurable for all k. Now the measures / i i , . . . ,/ifc,...
induce two-valued measures ^ , . . . , / ^ , . . . on X\(Mi UM
). By the Gitik-Shelah theorem,
there exists a set M3 C X\(M\ U M
) which is ^-nonmeasurable for all k. Continuing
in this way, we get a sequence of pairwise disjoint sets M i , . . . , M*,.. . such that Mk is
a /u-nonmeasurable set (in fact, for every n, Mn is a //fc-nonmeasurable set for all k).
On the other hand, let M i , . . . , M/t,... be pairwise disjoint sets, and let Mk be a //*-
nonmeasurable set. For every n consider disjoint ^
-nonmeasurable sets M'n, M^ such
that Mn = M'n U M". Clearly, |J
M'n is a ^^-nonmeasurable set for all k. D
The author does not consider himself competent to describe the entire history of the
main topic of this memoir. Suffice it to say that between the publication of Erdos' paper
[Er] and Shelah's paper [S] there appeared perhaps two dozen publications in which the
ideas of Ulam, Alaoglu and Erdos were developed. One of the more recent papers is
that of Grzegorek [G] in which, as stated by the author in his abstract, he proves a
theorem generalizing results of Ulam, Alaoglu-Erdos, Jensen, Prikry and Taylor connected
with Ulam's problem about sets of measures. Grzegorek's theorem implies the following
Corollary. Let 5 be a family of a-helds on the real line R such that for every A G 5
all one-element subsets of R belong to A and A / ^(R)- Then any of the conditions
(i) |5| w; (ii) |3 | 2^ and 2U = ui; (Hi) |#| 22" and GodeVs axiom of constructibility;
implies US ^V(R).
Claims 1.1 and 1.2, as stated above, are significant generalizations of Grzegorek's result
in respect to conditions (i) and (ii).
One usually says that the measure fij, induces a measure jxj, on the set X\M\. Obviously, /ifc will
induce a measure on a set L C A' if and only if Hk{L) ^ 0.
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