To expand on the answer of Carl Mummert, there is an increadibly useful theorem due to Kuratowski that says the following: There exists a measurable bijection with measurable inverse between any two Polish spaces of the same cardinality and this cardinality is either (at most) countable or the cardinality of the continuum. It follows that, as a measurable space, the Cantor set is really the same space a $\mathbb{R}$, $[0,1]$, or $[0,1]^\infty$. So the uniform distribution on $[0,1]$ corresponds to some atomless probability measure on the Cantor set. Now there is a result that says that every finite measure on the Borel sets of a metrizable metric space is automatically regular, so requiring regularity doesn't really change the problem.