To deal with infinite , we will now extend the treatment of semimeasures on to nontraditional probability distributions on .

Unfortunately it turns out that the analogue of the universal CEM Theorem 4.1 for EOMs with random input does not hold for GTMs:

**Proof.** The following proof is due to M. Hutter
(personal communications by email following a discussion of
approximable universes on 2 August 2000 in
Munich). It is an extension of a modified^{1} proof [30, p. 249 ff]
that there is no universal recursive semimeasure.

It suffices to focus on . Identify strings with
integers, and assume is a universal approximable
semidistribution. We construct an approximable semidistribution
that is not dominated by , thus contradicting the
assumption. Let
be a sequence of
recursive functions converging to . We recursively define a
sequence
converging to . The basic
idea is: each contribution to is the sum of consecutive
probabilities ( increasing). Define ;
. Let be such that
. Define as the element with
smallest (largest ) probability in this interval,
i.e.,
. If
is less than twice and
is more than half of
, set
. Otherwise set
for and for . is
obviously total recursive and non-negative. Since
, we have

Summing over we observe that if is a semidistribution, so is . From some on, changes by less than a factor of 2 since converges to . Hence remains unchanged for and converges to . But , violating our universality assumption .