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## Universal TM-Induced Measures

Definition 4.16 (P-Induced Measure )   Given a distribution on , define a measure on as follows:
 (28)

Note that (compare Def. 4.1):
 (29)

For those without 0-bit we have , for the others
 (30)

Definition 4.17 (TM-Induced Semimeasures )   Given some TM , for define . Again we deviate a bit from Levin's -oriented path [45] (survey: [30, p. 245 ff, p. 272 ff]) and extend to , where we define . If denotes a set of TMs with universal element , then we write
 (31)

We observe that is universal among all T-induced semimeasures, . Note that
 (32)

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

Corollary 4.2   For , is a CEM and approximable as the difference of two c.e. values: for without any 0-bit, otherwise
 (33)

Next: Universal CEM vs EOM Up: Measures and Probability Distributions Previous: TM-Induced Distributions and Convergence
Juergen Schmidhuber 2003-02-13