  Bennett's notion of depth is usually considered to describe the usefulness
and internal organization of the information encoded into an object such as an
infinite binary sequence. We consider a natural way to relativize the notion of
depth for such sets, and we investigate for various kinds of oracles whether
and how the unrelativized and the relativized version of depth differ.
Intuitively speaking, access to an oracle increases computation power.
Accordingly, for most notions for sets considered in computability theory, for
the corresponding classes trivially for all oracles the unrelativized class is
contained in the relativized class or for all oracles the relativized class is
contained in the unrelativized class. Examples for these two cases are given by
the classes of computable and of Martin-L\"{o}f random sets, respectively.
However, in the case for depth the situation is different. It turns out that
the classes of deep sets and of sets that are deep relative to the halting set
$\emptyset '$ are incomparable with respect to set-theoretical inclusion. On
the other hand, the class of deep sets is strictly contained in the class of
sets that are deep relative to any given Martin-L\"{o}f-random oracle. The set
built in the proof of the latter result can also be used to give a short proof
of the known fact that every PA-complete degree is Turing-equivalent to the
join of two Martin-L\"{o}f-random sets. In fact, we slightly strengthen this
result by showing that every DNC$_2$ function is truth-table-equivalent to the
join of two Martin-L\"{o}f random sets. Furthermore, we observe that the class
of deep sets relative to any given K-trivial oracle either is the same as or is
strictly contained in the class of deep sets. Obviously, the former case
applies to computable oracles. We leave it as an open problem which of the two
possibilities can occur for noncomputable K-trivial oracles.
