  We construct a family of non-parametric (infinite-dimensional) manifolds of
finite measures on $R^d$. The manifolds are modelled on a variety of weighted
Sobolev spaces, including Hilbert-Sobolev spaces and mixed-norm spaces. Each
supports the Fisher-Rao metric as a weak Riemannian metric. Densities are
expressed in terms of a deformed exponential function having linear growth.
Unusually for the Sobolev context, and as a consequence of its linear growth,
this "lifts" to a nonlinear superposition (Nemytskii) operator that acts
continuously on a particular class of mixed-norm model spaces, and on the fixed
norm space $W^{2,1}$; i.e. it maps each of these spaces continuously into
itself. It also maps continuously between other fixed-norm spaces with a loss
of Lebesgue exponent that increases with the number of derivatives. Some of the
results make essential use of a log-Sobolev embedding theorem. Each manifold
contains a smoothly embedded submanifold of probability measures. Applications
to the stochastic partial differential equations of nonlinear filtering (and
hence to the Fokker-Planck equation) are outlined.
