2 L.S. GRINBLAT
(b) |X\UnM^|K0.
The existence of Ulam's matrix implies the following important corollary:
Corollary. Let (i be a nontrivial a-additive measure defined on a set of cardinality Ki
("nontrivial" means that /JL(M) 0 for some M) such that fi({x}) = 0 for every singleton
{x}. Then there exists &i pairwise disjoint fi-nonmeasurable sets.
This property of the cardinality fc^ implies a well-known theorem of Alaoglu and Erdos,
proved in [Er]:
Theorem. Let / i i , . . . ,/4fc,... be a countable sequence of two-valued measures1 defined
on a set of cardinality Ni. Then there exists a set which is nonmeasurable relative to all
these measures.
We might mention that the special case of the Alaoglu-Erdos theorem for a finite se-
quence of measures was proved by Ulam. Moreover, the assumption that the measures are
two-valued is not essential. The existence of Ulam's matrix implies that the Alaoglu-Erdos
theorem remains in force provided that the measures /ik are nontrivial, a-additive, and
that for every singleton {x}, fJk({x}) = 0.
Alaoglu and Erdos arrived at their theorem as a result of their work on the following
problem of Ulam:
Problem. Find the minimal cardinal k such that, for any family of less than k two-valued
measures defined on a set of cardinality Ni, there exists a set which is nonmeasurable
relative to all these measures.
Ulam's problem was solved by Shelah (see[S]), who constructed a model in which k = Ki.
In Godel's model L, however, k K2.
We now proceed to formulate the main subject of this memoir. Our interest will be
focused on sets which are not members of algebras of sets. Unless otherwise stated, all
algebras here are defined on a certain abstract set X of arbitrary cardinality. By an algebra
A we mean a collection of subsets of X possessing the following property:
1
A measure n defined on a set X is said to be two-valued if (1) /x is (r-additive, fi(X) = 1, and fi({x}) = 0
for x £ X; (2) if M is a /i-measurable set, then either ft(M) = 0 o r /i(M) = 1.
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