  We investigate which infinite binary sequences (reals) are effectively random
with respect to some continuous (i.e., non-atomic) probability measure. We
prove that for every n, all but countably many reals are n-random for such a
measure, where n indicates the arithmetical complexity of the Martin-L\"of
tests allowed. The proof is based on a Borel determinacy argument and
presupposes the existence of infinitely many iterates of the power set of the
natural numbers. In the second part of the paper we present a metamathematical
analysis showing that this assumption is indeed necessary. More precisely,
there exists a computable function G such that, for any n, the statement `All
but countably many reals are G(n)-random with respect to a continuous
probability measure' cannot be proved in $ZFC^-_n$. Here $ZFC^-_n$ stands for
Zermelo-Fraenkel set theory with the Axiom of Choice, where the Power Set Axiom
is replaced by the existence of n-many iterates of the power set of the natural
numbers. The proof of the latter fact rests on a very general obstruction to
randomness, namely the presence of an internal definability structure.
