Universal TM-Induced Measures
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## Universal TM-Induced Measures

Definition 4.16 (P-Induced Measure tex2html_wrap_inline$P$)   Given a distribution P on , define a measure on B* as follows:

 (30)

Note that (compare Def. 4.1):

 (31)

For those without 0-bit we have , for the others

 (32)

Definition 4.17 (TM-Induced Semimeasures tex2html_wrap_inline$_T,^M,^E,^G$)   Given some TM T, for define . Again we deviate a bit from Levin's B*-oriented path [#!Zvonkin:70!#] (survey: [#!LiVitanyi:97!#, p. 245 ff, p. 272 ff]) and extend to , where we define . If C denotes a set of TMs with universal element UC, then we write

 (33)

We observe that is universal among all T-induced semimeasures, . Note that

 (34)

It will be obvious from the context when we deal with the restriction of to B*.

Corollary 4.2   For , is a CEM and approximable as the difference of two enumerable values: for x without any 0-bit, otherwise

 (35)

Next: Universal CEM vs EOM Up: Measures and Probability Distributions Previous: TM-Induced Distributions and Convergence
Juergen Schmidhuber
2001-01-09

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