Measurability in Mass Transportation and Harmonic Analysis

It is a recurring phenomenon in analysis that in the presence of a compatible additional (topological or measurable) structure algebraic invariance properties are so strong, that they already imply regularity. One of the most prominent examples of this principle is the work of D. Montgomery and L. Zippin regarding Hilbert`s 5th problem: continuity + algebraic invariance implies analyticity. A classical, but more elementary example of the implication measurability + algebraic invariance implies regularity} is the n-dimensional Cauchy functional equation f(x+y)=f(x)+f(y), the solutions of which are linear and therefore are even analytic. - Mass Transport: We want to investigate the role of measurability in the Monge-Kantorovich problem of optimal transportation; in characterizing optimality of transport plans the concept of c-monotonicity turned out to be crucial and it is subject to ongoing research to find equivalent formulations based solely on measurability conditions. Also questions regarding the Monge-Kantorovich Duality, the existence and uniqueness of primal and dual optimizers should be treated under this viewpoint. Besides methods from measure theory also results from descriptive set theory come into play. Techniques from mass transportation may be used to reformulate and treat problems in uniform distribution and convex geometry, which will be an important part of the planned research. - Hartman measurability: Hartman functions arise in the theory of uniform distribution on abstract groups where one wants to adapt the notion of Riemann integrability. Hartman functions are precisely the measurable functions with respect to a certain Boolean algebra which is closely related to continuity-sets and therefore share remarkable regularity properties. Concerning the structure of Hartman functions, however, not much is known and results concerning the reconstruction via Fourier data, resp. the Hartman measurability of Fourier transforms of measures are of particular interest. The dynamical systems associated to the translation of Hartman functions and, more general, to so called Banach almost periodic functions, are relevant to infinite combinatorics.