Corollary 4.3 and Lemma 4.2 below imply that and are essentially the same thing: randomly selecting the inputs of a universal EOM yields output prefixes whose probabilities are determined by the universal CEM.
T applies EOM0 to all
in dovetail fashion,
simply reads randomly selected input bits forever.
At a given time, let string variable z denote T's input
string read so far.
Starting at the right end of the unit interval [0,1),
are being updated by the algorithm of
Theorem 4.1, T
keeps updating a chain of finitely many, variable, disjoint,
consecutive, adjacent, half-open intervals
in alphabetic order on x,
such that VI(y) is to the right of VI(x) if .
After every variable update and each increase of
z, T replaces its output by the
x of the VI(x) with
Since neither z nor the
in the algorithm of Theorem 4.1
can decrease (that is, all interval boundaries can only shift left),
T's output cannot either, and therefore is indeed EOM-computable.
Obviously the following holds: