  The paper considers quantitative versions of different randomness notions:
algorithmic test measures the amount of non-randomness (and is infinite for
non-random sequences). We start with computable measures on Cantor space (and
Martin-Lof randomness), then consider uniform randomness (test is a function of
a sequence and a measure, not necessarily computable) and arbitrary
constructive metric spaces. We also consider tests for classes of measures, in
particular Bernoulli measures on Cantor space, and show how they are related to
uniform tests and original Martin-Lof definition. We show that Hyppocratic
(blind, oracle-free) randomness is equivalent to uniform randomness for
measures in an effectively orthogonal effectively compact class. We also
consider the notions of sparse set and on-line randomness and show how they can
be expressed in our framework.
