I have pretty no knowledge in set theory, so likely the question has a trivial answer. All countable subsets of $[0,1]$ have Lebesgue measure of zero, thus all sets of positive Lebesgue measure are uncountable. Does it yet mean that all these sets have same cardinality as $[0,1]$? Clearly, the answer is yes under the continuum hypothesis, but I wonder whether CH is crucial here and what would be the answer without CH.
I guess there is no difference whether we consider only Borel sets, or all Lebesgue measurable ones.