##
Finite Self-Information

Status: published in *Computability* 1
(2012) 85 - 98.

Availability: journal
version and preprint

**Abstract.** We present a definition, due to Levin, of mutual
information *I*(*A*:*B*) for infinite sequences. We
say that a set *A* *has finite self-information* if
*I(A:A)* < ∞. It is easy to see that every *K*-trivial
set has finite self-information. We answer a question of Levin by showing
that the converse does not hold. Finally, we investigate the
connections between having finite self-information and other notions of
weakness such as jump-traceability. In particular, we show that our
proof can be adapted to produce a set that is low for both effective
Hausdorff dimension and effective packing dimension, but
not *K*-trivial.

drh@math.uchicago.edu