Continuous randomness via transformations of 2-random sequences

Reimann and Slaman initiated the study of sequences that are Martin-Löf random with respect to a continuous measure. In the case of sequences that are random with respect to a computable, continuous measure, the picture is understood: such sequences are truth-table equivalent to a Martin-Löf random sequence. However, we may ask: Given a sequence that is random with respect to a continuous measure but not with respect to any computable measure, how close to effective is the measure with respect to which it is continuously random? In this study, we take up this question by examining various transformations of 2-random sequences to establish several results on sequences that are continuously random with respect to a measure that is computable in ∅′ but not random with respect to a computable measure. In addition, we obtain similar results when transforming 2-randomness under a wider class of effective operators.