  In persistent topology, q-tame modules appear as a natural and large class of
persistence modules indexed over the real line for which a persistence diagram
is definable. However, unlike persistence modules indexed over a totally
ordered finite set or the natural numbers, such diagrams do not provide a
complete invariant of q-tame modules. The purpose of this paper is to show that
the category of persistence modules can be adjusted to overcome this issue. We
introduce the observable category of persistence modules: a localization of the
usual category, in which the classical properties of q-tame modules still hold
but where the persistence diagram is a complete isomorphism invariant and all
q-tame modules admit an interval decomposition.
