  Reimann and Slaman initiated the study of sequences that are Martin-L\"of
random with respect to a continuous measure, establishing fundamental facts
about NCR, the collection of sequences that are not Martin-L\"of random with
respect to any continuous measure. In the case of sequences that are random
with respect to a computable, continuous measure, the picture is fairly
well-understood: such sequences are truth-table equivalent to a Martin-L\"of
random sequence. However, given a sequence that is random with respect to a
continuous measure but not with respect to any computable measure, we can ask:
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 (sequences that are Martin-L\"of random relative to the
halting set $\emptyset'$) to establish several results on sequences that are
continuously random with respect to a measure that is computable in
$\emptyset'$. In particular, we show that (i) every noncomputable sequence that
is computable from a 2-random sequence is Martin-L\"of random with respect to a
continuous, $\emptyset'$-computable measure and (ii) the Turing jump of every
2-random sequence is Martin-L\"of random with respect to a continuous,
$\emptyset'$-computable measure. From these results, we obtain examples of
sequences that are not proper, i.e., not random with respect to any computable
measure, but are random with respect to a continuous, $\emptyset'$-computable
measure. Lastly, we consider the behavior of 2-randomness under a wider class
of effective operators (c.e. operators, pseudojump operators, and operators
defined in terms of pseudojump inversion), showing that these too yield
sequences that are Martin-L\"of random with respect to a continuous,
$\emptyset'$-computable measure.
