TITLE: Do sets with positive Lebesgue measure have same cardinality as R?
QUESTION [38 upvotes]: I have been thinking about which kind of wild non-measurable functions you can define. This led me to the question:
Is it possible to prove in ZFC, that if a (Edit: measurabel) set $A\subset \mathbb{R}$ has positive Lebesgue-measure, it has the same cardinality as $\mathbb{R}$? It is obvious if you assume CH, but can you prove it without CH?

REPLY [10 votes]: A set of positive measure contains a closed subset of positive measure. It is known that closed subset of reals may have cardinality either continuum or at most countable
(http://en.wikipedia.org/wiki/Perfect_set_property)