Non-atomic doubly stochastic measures
Permutation matrices have played special roles in the class of doubly stochastic matrices. Generalizing to doubly stochastic measures, recently-defined non-atomic doubly stochastic measures (DSMs) can be viewed as a continuous analog of permutation matrices. We investigate basic properties of the class of non-atomic DSMs, e.g.~its relationship with other well-known subclasses of the DSMs and their partial factorizability with respect to the product induced by the isomorphism to the class of Markov operators endowed with the composition operator. We then obtain a characterization of completely factorizable DSMs as well as a procedure to determine factorizability. As examples, we consider DSMs which are fixed points of some iterated function systems and DSMs supported on a hairpin.