  We study the question, ``For which reals $x$ does there exist a measure $\mu$
such that $x$ is random relative to $\mu$?'' We show that for every
nonrecursive $x$, there is a measure which makes $x$ random without
concentrating on $x$. We give several conditions on $x$ equivalent to there
being continuous measure which makes $x$ random. We show that for all but
countably many reals $x$ these conditions apply, so there is a continuous
measure which makes $x$ random. There is a meta-mathematical aspect of this
investigation. As one requires higher arithmetic levels in the degree of
randomness, one must make use of more iterates of the power set of the
continuum to show that for all but countably many $x$'s there is a continuous
$\mu$ which makes $x$ random to that degree.
