Rectangle Measures

This chapter develops the theory of rectangle measures: finitely-additive tiling measures defined on rectangles in the plane. Every real-parameter persistence module gives rise to such a measure, the ‘persistence measure’ of the module. An equivalence theorem asserts that a rectangle measure can be represented as a diagram of decorated points in the plane. In particular, the persistence measure of a persistence module gives rise to its persistence diagram. The diagram carries no structural information in regions of the plane where the measure is infinite. For this reason, we isolate various tameness conditions on persistence modules that guarantee finiteness in regions of the extended plane; the most important of these is q-tameness. Vanishing lemmas ease the computation of persistence diagrams by identifying regions of the plane where the diagram is empty. Finally, we show that our measure-theoretic diagrams agree with the traditionally defined diagrams in certain standard settings (such as the sublevelset persistent homology of a Morse function on a compact manifold).