Dixmier measurability in Marcinkiewicz spaces and applications

We present new results concerning the description of Dixmier measurable elements in the sense of A. Connes in Marcinkiewicz spaces and demonstrate some applications of our results. In particular, we complement famous Connes Trace Theorem by describing the asymptotics of the singular function of order −n classical pseudo-differential operators on an n-dimensional compact Riemannian manifold.