  We examine a definition of the mutual information of two reals proposed by
Levin. The mutual information is
I(X:Y)=log(sum(2^{K(s)-K^X(s)+K(t)-K^Y(t)-K(s,t)}, (s,t) pairs of finite binary
strings), where K is the prefix-free Kolmogorov complexity. A real X is said to
have finite self-information if I(X:X) is finite. We give a construction for a
perfect Pi^0_1 set of reals with this property, which settles some open
questions posed by Hirschfeldt and Weber. The construction produces a perfect
set of reals with K(s)<= K^{A}(s)+f(s)+c for a certain constant c and for any
given Delta^0_2 f with a particularly nice approximation and for a specific
choice of f it can also be used to produce a perfect Pi^0_1 set of reals that
are low for effective Hausdorff dimension and effective packing dimension. The
construction can be further adapted to produce a single perfect set of reals
that satisfy K(s) <= K^A(s)+f(s)+c_f, where c_f is a constant that depends on
f, for all f in a `nice' class of Delta^0_2 functions which includes all
recursive orders.
