Suppose TM 's input bits are obtained by tossing an unbiased coin
whenever a new one is requested.
Levin's *universal discrete enumerable semimeasure*
[28,14,16]
or *semidistribution*
is limited to and halting programs:

Note that if universal. We will now generalize this in obvious but nontraditional ways to and nonhalting programs for MTMs, EOMs, and GTMs.

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would be

For notational simplicity, however, we will continue using the sign to indicate summation over uncountable objects, rather than using a measure-oriented notation for probability densities. The reader should not worry about this -- the theorems in the remainder of the paper will focus on those with ; density-like nonzero sums over uncountably many bitstrings, each with individual measure zero, will not play any critical role in the proofs.

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Since all programs of EOMs and MTMs
converge, and are proper probability distributions on .
For instance,
.
, however, is just a semidistribution.
To obtain a proper probability
distribution , one might think of normalizing
by the *convergence probability* :

Even and are generally not describable for , in the sense that there is no GTM that takes as an input a finite description (or program) of any M-describable or E-describable and converges towards or . This is because in general it is not even weakly decidable (Def. 2.11) whether two programs compute the same output. If we know that one of the program outputs is finite, however, then the conditions of weak decidability are fulfilled. Hence certain TM-induced distributions on are describable, as will be seen next.

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**Proof.** The following algorithm computes (compare
proof of Theorem 3.3):

Initialize the real-valued variable by 0, run all possible programs of EOM dovetail style; whenever the output of a program prefix starts with some for the first time, set ; henceforth ignore continuations of .In this way enumerates . Infinite are not problematic as only a finite prefix of must be read to establish if the latter indeed holds.

Similarly, facts of the form can be discovered after finite time.

Now we will make the connection to the previous subsection on semimeasures on .