The problem with lower semicomputable functions is that it’s a class not closed under natural operations. For example, taking minus such a function we get an upper semicomputable function that can fail to be lower semicomputable. So, given a Solomonoff induction oracle we can very easily (i.e. using a very efficient oracle machine) construct measures that are not absolutely continuous w.r.t. the Solomonoff prior.
In fact, for any prior this can be achieved by constructing an “anti-inductive” sequence: a sequence that contains 1 at a given place if and only if the prior, conditional on the sequence before this place, assigns probability less than 12 to 1. Such a sequence cannot be accurately predicted by the prior (and, by the merging-of-opinions theorem, a delta-function at this sequence it is not absolutely continuous w.r.t. the prior).
The problem with lower semicomputable functions is that it’s a class not closed under natural operations. For example, taking minus such a function we get an upper semicomputable function that can fail to be lower semicomputable. So, given a Solomonoff induction oracle we can very easily (i.e. using a very efficient oracle machine) construct measures that are not absolutely continuous w.r.t. the Solomonoff prior.
In fact, for any prior this can be achieved by constructing an “anti-inductive” sequence: a sequence that contains 1 at a given place if and only if the prior, conditional on the sequence before this place, assigns probability less than 12 to 1. Such a sequence cannot be accurately predicted by the prior (and, by the merging-of-opinions theorem, a delta-function at this sequence it is not absolutely continuous w.r.t. the prior).