  We study the continuity and the measurability of the solution to
Schr\"odinger's functional equation, with respect to space, kernel and
marginals, provided the space of all Borel probability measures is endowed with
the weak topology. This is a continuation of our previous result where the
space of all Borel probability measures was endowed with the strong topology.
As an application, we construct a convex function of which the moment measure
is a given probability measure, by the zero noise limit of a class of
stochastic optimal transportation problems.
