  We study the possible growth rates of the Kolmogorov complexity of initial
segments of sequences that are random with respect to some computable measure
on $2^\omega$, the so-called proper sequences. Our main results are as follows:
(1) We show that the initial segment complexity of a proper sequence $X$ is
bounded from below by a computable function (that is, $X$ is complex) if and
only if $X$ is random with respect to some computable, continuous measure. (2)
We prove that a uniform version of the previous result fails to hold: there is
a family of complex sequences that are random with respect to a single
computable measure such that for every computable, continuous measure $\mu$,
some sequence in this family fails to be random with respect to $\mu$. (3) We
show that there are proper sequences with extremely slow-growing initial
segment complexity, that is, there is a proper sequence the initial segment
complexity of which is infinitely often below every computable function, and
even a proper sequence the initial segment complexity of which is dominated by
all computable functions. (4) We prove various facts about the Turing degrees
of such sequences and show that they are useful in the study of certain classes
of pathological measures on $2^\omega$, namely diminutive measures and trivial
measures.
