The next three definitions concerning semimeasures on B^{*} are almost
but not quite identical to those of discrete semimeasures
[#!LiVitanyi:97!#, p. 245 ff]
and continuous semimeasures
[#!LiVitanyi:97!#, p. 272 ff]
based on the work of Levin and Zvonkin
[#!Zvonkin:70!#].

Definition 4.1 (Semimeasures)
A (binary) semimeasure
is a function
that satisfies:

(17)

where
is a function
satisfying
.

A notational difference to the approach of Levin
[#!Zvonkin:70!#]
(who writes
)
is the explicit introduction of .
Compare the
introduction of an undefined element u by Li and
Vitanyi [#!LiVitanyi:97!#, p. 281].
Note that
.
Later we will discuss the interesting case
,
the a priori probability of x.

Definition 4.2 (Dominant Semimeasures)
A semimeasure
dominates another semimeasure
if
there is a constant
such that for all

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Definition 4.3 (Universal Semimeasures)
Let
be a set of semimeasures on B^{*}.
A semimeasure
is universal if it
dominates all
.

In what follows,
we will introduce describable semimeasures dominating those considered
in previous work ([#!Zvonkin:70!#], [#!LiVitanyi:97!#, p. 245 ff, p.272 ff]).