  By a theorem of Sacks, if a real $x$ is recursive relative to all elements of
a set of positive Lebesgue measure, $x$ is recursive. This statement, and the
analogous statement for non-meagerness instead of positive Lebesgue measure,
have been shown to carry over to many models of transfinite computations. Here,
we start exploring another analogue concerning recognizability rather than
computability. We introduce a notion of relativized recognizability and show
that, for Infinite Time Turing Machines (ITTMs), if a real $x$ is recognizable
relative to all elements of a non-meager Borel set $Y$, then $x$ is
recognizable. We also show that a relativized version of this statement holds
for Infinite Time Register Machines (ITRMs). This extends our earlier work
where we obtained the (unrelativized) result for ITRMs. We then introduce a
jump operator for recognizability, examine its set-theoretical content and show
that the recognizable jumps for ITRMs and ITTMs are primitive-recursively
equivalent, even though these two models are otherwise of vastly different
strength. Finally, we introduce degrees of recognizability by considering the
transitive closure of relativized recognizability and connect it with the
recognizable jump operator to obtain a solution to Post's problem for degrees
of recognizability.
