ﻻ يوجد ملخص باللغة العربية
We study degree-theoretic properties of reals that are not random with respect to any continuous probability measure (NCR). To this end, we introduce a family of generalized Hausdorff measures based on the iterates of the dissipation function of a continuous measure and study the effective nullsets given by the corresponding Solovay tests. We introduce two constructions that preserve non-randomness with respect to a given continuous measure. This enables us to prove the existence of NCR reals in a number of Turing degrees. In particular, we show that every $Delta^0_2$-degree contains an NCR element.
We investigate which infinite binary sequences (reals) are effectively random with respect to some continuous (i.e., non-atomic) probability measure. We prove that for every n, all but countably many reals are n-random for such a measure, where n ind
The Posner-Robinson Theorem states that for any reals $Z$ and $A$ such that $Z oplus 0 leq_mathrm{T} A$ and $0 <_mathrm{T} Z$, there exists $B$ such that $A equiv_mathrm{T} B equiv_mathrm{T} B oplus Z equiv_mathrm{T} B oplus 0$. Consequently, any non
We study pairs of reals that are mutually Martin-L{o}f random with respect to a common, not necessarily computable probability measure. We show that a generalized version of van Lambalgens Theorem holds for non-computable probability measures, too. W
We define the bounded jump of A by A^b = {x | Exists i <= x [phi_i (x) converges and Phi_x^[A|phi_i(x)](x) converges} and let A^[nb] denote the n-th bounded jump. We demonstrate several properties of the bounded jump, including that it is strictly in
We study the question, ``For which reals $x$ does there exist a measure $mu$ such that $x$ is random relative to $mu$? We show that for every nonrecursive $x$, there is a measure which makes $x$ random without concentrating on $x$. We give several co