Uniqueness of Banach space valued graphons

A Banach space valued graphon is a function W : (ohm, A, pi)(2) -> Z from a probability space to a Banach space with a separable predual, measurable in a suitable sense, and lying in appropriate L-P-spaces. As such we may consider W(x, y) as a two-variable random element of the Banach space. A two-dimensional analogue of moments can be defined with the help of graphs and weak-* evaluations, and a natural question that then arises is whether these generalized moments determine the function W uniquely - up to measure preserving transformations. The main motivation comes from the theory of multigraph limits, where these graphons arise as the natural limit objects for convergence in a generalized homomorphism sense. Our main result is that this holds true under some Carleman-type condition, but fails in general even with Z = R, for reasons related to the classical moment-problem. In particular, limits of multigraph sequences are uniquely determined - up to measure preserving transformations - whenever the tails of the edge-distributions stay small enough. (C) 2019 Elsevier Inc. All rights reserved.