  We present a unified approach to the study of Hilbert-Kunz multiplicity,
F-signature, and related limits governed by Frobenius and Cartier linear
actions in positive characteristic commutative algebra. We introduce general
techniques that give vastly simplified proofs of existence, semicontinuity, and
positivity. Furthermore, we give an affirmative answer to a question of
Watanabe and Yoshida allowing the F-signature to be viewed as the infimum of
relative differences in the Hilbert-Kunz multiplicites of the cofinite ideals
in a local ring.
