  We formulate conditions for the solvability of the problem of robust utility
maximization from final wealth in continuous time financial markets, without
assuming weak compactness of the densities of the uncertainty set, as customary
in the literature. Relevant examples of such a situation typically arise when
the uncertainty set is determined through moment constraints. Our approach is
based on identifying functional spaces naturally associated with the elements
of each problem. For general markets these are modular spaces, through which we
can prove a minimax equality and the existence of optimal strategies by
exploiting the compactness, which we establish, of the image by the utility
function of the set of attainable wealths. In complete markets we obtain
additionally the existence of a worst-case measure, and combining our ideas
with abstract entropy minimization techniques, we moreover provide in that case
a novel methodology for the characterization of such measures.
