In 2008 Jan Reimann and Ted Slaman showed that every non-recursive real has a measure such that the real is not an atom of the measure and such that the real is Martin-Löf random relative to the measure. In this talk I will explore the question of whether or not this result can be raised to notions of randomness stronger than Martin-Löf, and discuss, relative to stronger notions of randomness, the problem of characterizing the set of reals for which such measures exist.