The main result of the paper establishes the compatibility of inductive limits of (topological) measure spaces with \(L^p\)-space theory, by showing the isometry of the Banach space \(L^p\) of an inductive limit of measure spaces with the projective limit of the \(L^p\)-spaces of the factors in the inductive limit for \(1\leq p\leq\infty\). This sort of problems has been neglected a long time and the paper makes clear why such problems couldn't be treated successfully as yet: There was no appropriate notion of morphism for the category of (topological) measure spaces available. It is the merit of the present paper that it introduces new morphisms which are defined as equivalence classes of measurable, measure reducing mappings under the equivalence relation ``inverse images of sets differ at most on a set of measure zero'', thus providing a category for establishing the main result. The above equivalence relation has been already used in stochastic processes and in ergodic theory in a completely different context. The main tool in the proof is Graf's result on the point-realization of \(\sigma\)-homomorphisms.