It is a result of Levin and (independently?) Kautz that if $X \in 2^\omega$ is 1-ML random relative to a computable measure $\mu$ the either $X$ is computable (it is an atom of $\mu$) or $X$ is Turing equivalent to a 1-ML random – and for computable continuous measures there is a computable measure preserving functional which preseves 1-ML randomness. Are other computable measures really totally trivial or is there the potential for interesting results if we relax some of those assumptions? In particular are there known answers to the following:
Is every 1-Schnor random with respect to a computable measure $\mu$ (using any reasonable generalization) of the same Turing degree as a 1-Schnor random? What about KL-randoms?
What if $\mu$ is a computable probability (or what about just $\sigma$-finite) measure on $\omega^\omega$?
Is there a reasonable class of measures -- for instance maybe those where the measure of each basic open set is 0' computable -- such that every real is random with respect to some measure in this class. I know that for continuous measures Slaman and Riemann have some great results about there only being countably many reals not n-random with respect to some arbitrary continuous measure and obviously if you don't restrict the class of measures at all you can just make any set an atom. However, my intuition says that if you somehow make the class of measures sufficiently similar to the tests being used (eg so a failed test provides a measure that concentrates more on the set) then every set should be random with respect to some measure in the class.
What happens if you move to $\Delta_1^1$ or $\Pi^1_1$ tests/null classes and consider hyperarithmetic measures? Do those also offer nothing new of interest relative to the standard Lebesgue measure?
In short, is there any interesting reason to look at other measures in randomness aside from Slaman and Riemann style results about continuous measures?
