  A basic information theoretic model for summarization is formulated. Here
summarization is considered as the process of taking a report of $v$ binary
objects, and producing from it a $j$ element subset that captures most of the
important features of the original report, with importance being defined via an
arbitrary set function endemic to the model. The loss of information is then
measured by a weight average of variational distances, which we term the
semantic loss.
  Our results include both cases where the probability distribution generating
the $v$-length reports are known and unknown. In the case where it is known,
our results demonstrate how to construct summarizers which minimize the
semantic loss. For the case where the probability distribution is unknown, we
show how to construct summarizers whose semantic loss when averaged uniformly
over all possible distribution converges to the minimum.
