  Let $R$ be an excellent Noetherian ring of prime characteristic. Consider an
arbitrary nested pair of ideals (or more generally, a nested pair of submodules
of a fixed finite module). We do \emph{not} assume that their quotient has
finite length. In this paper, we develop various sufficient numerical criteria
for when the tight closures of these ideals (or submodules) match. For some of
the criteria we only prove sufficiency, while some are shown to be equivalent
to the tight closures matching. We compare the various numerical measures (in
some cases demonstrating that the different measures give truly different
numerical results) and explore special cases where equivalence with matching
tight closure can be shown. All of our measures derive ultimately from
Hilbert-Kunz multiplicity.
