Consider the following question:
under what conditions on a collection of subsets of the unit interval can the existence
of an extension of Lebesgue measure defined on each element of the collection
be guaranteed? The main purpose of this paper is to find conditions on
the cardinality of the collection whose sufficiency can be shown consistent
without the use of large cardinals. For example, if ZFC is consistent so is
ZFC + “Lebesgue measure can be extended to any countable collection of
sets”.
The results of this paper complement work of earlier researchers. Banach and
Kuratowski showed that assuming the continuum hypothesis there is a countable
collection of sets of reals for which no extension exists. Solovay proved that an
extension of Lebesgue measure to all sets is equiconsistent with the existence of a
measurable cardinal.
