  We present a categorical viewpoint of probability measures by showing that a
probability measure can be viewed as a weakly averaging affine measurable
functional taking values in the unit interval which preserves limits. The
probability measures on a space are the elements of a submonad of a double
dualization monad on the category of measurable spaces into the unit interval,
and this monad is naturally isomorphic to the Giry monad. We show this submonad
is the codensity monad of a functor from the category of convex spaces to the
category of measurable spaces. A theorem proving the integral operator acting
on the space of measurable functions and the space of probability measures on
the domain space of those functions is given using the strong monad structure
of the Giry monad.
