The Giry monad
Let X be a measurable space. The collection of all measures on X again forms a measurable space. This construction forms a monad on measurable spaces and measurable functions, called the Giry monad.
Note that most sources use the term "Giry monad" for the restriction to probability measures. Here we include all measures on X.
Measurability structure on
measure: Measures are measurable w.r.t. all projections
Monadic join on
measure in the category of measurable spaces and measurable
Monadic bind on
measure, only works in the category of measurable spaces and measurable
functions. When the function
f is not measurable the result is not well defined.