Is the Lebesgue measure of a set that is not dense = 0?

Question

If I have a set that is not dense in any interval in $\mathbb{R}$, then can I say that it has zero Lebesgue measure? Or is there a counter example?

Answer 1

Please see the Wikipedia article on the Smith-Volterra-Cantor set. It is nowhere dense and has positive Lebesgue measure.

For more information, google "fat Cantor set." Indeed a search of MSE will give a number of discussions of such sets.

Answer 2

No, there are such sets with Lebesgue measure arbitrarily close to the full measure. For instance for every$~N>0$, the set of irrational numbers $x$ such that for no integer $m>N$ the first sequence of $m$ decimals of $x$ after the decimal point equals the sequence of $m$ decimals immediately following it, is easily seen to not be dense in any interval. But by increasing $N$ its measure approximates the full measure rapidly.