We give a self-contained treatment of the theory of persistence modules
indexed over the real line. We give new proofs of the standard results.
Persistence diagrams are constructed using measure theory. Linear algebra
lemmas are simplified using a new notation for calculations on quiver
representations. We show that the stringent finiteness conditions required by
traditional methods are not necessary to prove the existence and stability of
the persistence diagram. We introduce weaker hypotheses for taming persistence
modules, which are met in practice and are strong enough for the theory still
to work. The constructions and proofs enabled by our framework are, we claim,
cleaner and simpler.Comment: New version. We discuss in greater depth the interpolation lemma for
persistence module
