  The existence of a unique Augustin mean and its invariance under the Augustin
operator are established for arbitrary input distributions with finite Augustin
information for channels with countably generated output $\sigma$-algebras. The
existence is established by representing the conditional R\'enyi divergence as
a lower semicontinuous and convex functional in an appropriately chosen
uniformly convex space and then invoking the Banach--Saks property in
conjunction with the lower semicontinuity and the convexity. A new family of
operators is proposed to establish the invariance of the Augustin mean under
the Augustin operator for orders greater than one. Some members of this new
family strictly decrease the conditional R\'enyi divergence, when applied to
the second argument of the divergence, unless the second argument is a fixed
point of the Augustin operator.
