  Chaitin's number Omega is the halting probability of a universal prefix-free
machine, and although it depends on the underlying enumeration of prefix-free
machines, it is always Turing-complete. It can be observed, in fact, that for
every computably enumerable (c.e.) real, there exists a Turing functional via
which Omega computes it, and such that the number of bits of omega that are
needed for the computation of the first n bits of the given number (i.e. the
use on argument n) is bounded above by a computable function h(n) = n+o(n). We
characterise the asymptotic upper bounds on the use of Chaitin's omega in
oracle computations of halting probabilities (i.e. c.e. reals). We show that
the following two conditions are equivalent for any computable function h such
that h(n)-n is non-decreasing: (1) h(n)-n is an information content measure,
(2) for every c.e. real there exists a Turing functional via which omega
computes the real with use bounded by h. We also give a similar
characterisation with respect to computations of c.e. sets from Omega, by
showing that the following are equivalent for any computable non-decreasing
function g: (1) g is an information-content measure, (2) for every c.e. set A,
Omega computes A with use bounded by g. Further results and some connections
with Solovay functions are given.
