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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 everynonrecursive $x$, there is a measure which makes $x$ random withoutconcentrating on $x$. We give several conditions on $x$ equivalent to therebeing continuous measure which makes $x$ random. We show that for all butcountably many reals $x$ these conditions apply, so there is a continuousmeasure which makes $x$ random. There is a meta-mathematical aspect of thisinvestigation. As one requires higher arithmetic levels in the degree ofrandomness, one must make use of more iterates of the power set of thecontinuum to show that for all but countably many $x$'s there is a continuous$\mu$ which makes $x$ random to that degree.

Probability Measures and Effective Randomness