Wenn die Macht gnadig wird und herabkommt in's Sichibare:
Schonheit heisse ich solches Herabkornmen
Friedrich Nietzsche
1. INTRODUCTION
Without a doubt, after the construction of the classical (Lebesgue) theory of measure,
some of the most interesting problems in set theory are those concerning nonmeasurable
sets. Reflections on this range of problems led Banach to formulate the following general
problem:
Problem. Does there exist a real-valued function f, defined on all subsets of the interval
(0,1), satisfying the following conditions:
(1) there exists a set M such that f(M) 0;
(2) for every singleton { x }, /({#}) = 0;
(3) if{Mn} is a countable sequence of pairwise disjoint sets, then /((J Mn) = ^ f(Mn).
In 1930 Ulam proved the following theorem which partly solves this problem:
Theorem . On the assumption that the continuum hypothesis is true (2^° = ^i) , there
exists no function f satisfying the above conditions.
Only forty years later it was shown that one can construct a model in which there exists
a a-additive extension of Lebesgue measure to all subsets of (0,1). This remarkable result
is due to Solovay (see [So]). Ulam's result follows from his striking discovery, known as
Ulam's matrix (see [U]):
Ulam's matrix. If X is a set of cardinality N1? there exists a matrix of subsets of X:
(M\
Ml ... Ml ...\
[ M ? Ml ... Ml ...
I
M[l M2n
... Ml ... I
having Ko rows and Hi columns such that
(a) M
a
n nM^ = $ifa^ (J;
Received by editor March 26, 1990 and in revised form August 26, 1991
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