A curious example from statistical differential geometry

We consider an example of a family of probability measures on an infinite dimensional space which are mutually singular. Although the Fisher information metric and its variants are not available, it is shown that the parameter manifold has a natural differential structure that is non-Riemannian with nonzero curvature. It is also shown that there is no Riemannian metric compatible with the natural affine connection for which the curvature is not zero.