We develop a family of infinite-dimensional Riemannian manifolds of finite measures. The latter are defined on an underlying Banach space, and have densities with respect to a reference measure that are of class C_b^k. (The case k=\infty, in which the manifolds are modelled on Frechet space, is included.) The manifolds admit the Fisher-Rao metric and the full geometry of Amari's \alpha-covariant derivatives for all real \alpha. The subset of probability measures of each manifold is shown to be a C^\infty-embedded submanifold. This embedding, together with the affine charts it supplies, is a natural way of studying the dually flat geometry of statistical manifolds. The likelihood function associated with a finite sample is a continuous linear function on each of the manifolds constructed.
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