  The Giry monad on the category of measurable spaces sends a space to a space
of all probability measures on it. There is also a finitely additive Giry monad
in which probability measures are replaced by finitely additive probability
measures. We give a characterisation of both finitely and countably additive
probability measures in terms of integration operators giving a new description
of the Giry monads. This is then used to show that the Giry monads arise as the
codensity monads of forgetful functors from certain categories of convex sets
and affine maps to the category of measurable spaces.
