TITLE: Lebesgue measure zero set of cardinality $\mathfrak c$
QUESTION [6 upvotes]: Suppose $A\subset\mathbb  R$ is a Lebesgue measure zero set. Must $\mathbb R\setminus A$ has cardinality $\mathfrak c$?
If so, does there exist another Lebesgue measure zero set $B$ of cardinality $\mathfrak c$ such that $A\cap B=\emptyset$?

REPLY [7 votes]: For 1., notice that ${\bf R}\setminus A$ has positive measure, so by regularity of Lebesgue measure it contains a compact $K$ set of positive measure. But compact sets in Polish spaces are either countable of of cardinality $\mathfrak c$.
For 2., notice that the closed set $K$ mentioned previously has positive measure and can be chosen to be perfect. You can then perform a construction similar to the construction of the Cantor set on $K$ to obtain a subset of $K$ of cardinality $\mathfrak c$, homeomorphic with the Cantor set, with zero measure (just make sure that at each step you leave only a small enough, measure wise, part).