The structure and stability of persistence modules

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.

💡 Research Summary

The paper presents a self‑contained development of persistence modules indexed by the real line, deliberately avoiding the heavy finiteness assumptions that dominate classical treatments. It begins by recalling the standard definition of a persistence module V : ℝ → Vectₖ and the usual “tame” or “finite‑type” hypotheses, then points out concrete scenarios—such as modules with infinitely many critical values or infinite‑dimensional fibers—where those hypotheses break down. To overcome these limitations, the authors introduce a measure‑theoretic construction of persistence diagrams. For any rectangle I ⊂ ℝ² they define a measure μ_V(I) equal to the rank of the linear map V(a ≤ b) for the corresponding interval