Description
This paper bridges a segment of measure and category theory. Essentially, measure theory is used to build some categorical structures. Given a sigma finite algebra with measure u, the set of measures absolutely continuous to u is considered. The set of nonnegative measurable functions is partitioned with u by equal almost everywhere. By use of the Radon-Nikodym Theorem, a bijective mapping T between the equivalence classes of the partitioned functions and the set of measures absolutely continuous to u can be built. The set of all measures absolutely continuous to all sigma finite measures on the given algebra are considered. These can be partitioned by the equivalence relation of mutually absolutely continuous. The equivalence classes can be partially ordered and each gives rise to a category by considering the mappings T in a certain way. Further, there exist functors between each category if the equivalence classes are comparable in the partial ordering. If in the class of all of these functors, each is considered to be a mapping, a category under functor composition is formed. Finally some topological properties of finite signed measures are considered and T is found, in the proper topologies, to be continuous.
