We examine a definition of mutual information for reals due to Levin. This
definition induces an associated lowness notion of having finite
self-information, that is, mutual information with oneself. Hirschfeldt
and Weber proved that the set of reals with finite self-information
strictly contains the K-trivials. We show that it is in fact much larger,
constructing a perfect $Pi^0_1$ class of such reals. The proof technique
involves a certain more general class of weakenings of lowness for K, and
we discuss some results regarding these notions and some other
applications.