  We develop a family of infinite-dimensional Banach manifolds of measures on
an abstract measurable space, employing charts that are "balanced" between the
density and log-density functions. The manifolds,
$(\tilde{M}_{\lambda},\lambda\in [2,\infty))$, retain many of the features of
finite-dimensional information geometry; in particular, the
$\alpha$-divergences are of class $C^{\lceil\lambda\rceil-1}$, enabling the
definition of the Fisher metric and $\alpha$-derivatives of particular classes
of vector fields. Manifolds of probability measures, $(M_{\lambda},\lambda\in
[2,\infty))$, based on centred versions of the charts are shown to be
$C^{\lceil\lambda \rceil-1}$-embedded submanifolds of the
$\tilde{M}_{\lambda}$. The Fisher metric is a pseudo-Riemannian metric on
$\tilde{M}_{\lambda}$. However, when restricted to finite-dimensional embedded
submanifolds it becomes a Riemannian metric, allowing the full development of
the geometry of $\alpha$-covariant derivatives. $\tilde{M}_{\lambda}$ and
$M_{\lambda}$ provide natural settings for the study and comparison of
approximations to posterior distributions in problems of Bayesian estimation.
