nLab table of probability monads

Probability monad ($P$)	Category	Elements/points of $P X$	Extra structure of $P X$	$P$-Algebras	References
distribution monad (a.k.a. finitary Giry monad, convex combination monad)	Set	convex combinations or finitely-supported probability measures	(just a set)	convex spaces	Fritz '09, Jacobs '18
Giry monad	Meas	probability measures	initial σ-algebra of evaluation maps	Full characterization unknown. See also here.	Lawvere '62, Giry '80
Giry monad	Pol	Borel probability measures	initial topology of integration maps	Full characterization unknown. See also here.	Giry '80
Probability monad on QBS?	quasi-Borel spaces	Equivalence classes of random variables	quasi-Borel structure inherited by the cartesian closed structure	Full characterization unknown.	HKSY '17
Radon monad	Comp	Radon probability measures (or continuous valuations)		compact convex subsets of locally convex topological vector spaces	Swirszcz '74, Keimel '08
ordered Radon monad	CompOrd	Radon probability measures (or continuous valuations)	weak topology w.r.t. continuous functions, stochastic order	compact convex subsets of ordered? locally convex topological vector spaces	Swirszcz '74, Keimel '08
measure monad on Top	Top	τ-additive measures	A-topology, stochastic order	Full characterization unknown. See also here.	F-P-R '19
probabilistic powerdomain?	dcpo, continuous domains?	continuous valuations	stochastic order	abstract probabilistic domains? (continuous case)	J-P '89
extended probabilistic powerdomain	Top, stably compact spaces	continuous valuations	initial topology of evaluation maps, stochastic order	Full characterization unknown. Dedicated section here	Heckmann '96, A-J-K '04, GL-J '19, F-P-R '19
valuation monad on locales?	Loc	continuous valuations	initial topology of evaluation maps	…	Vickers '11
Kantorovich monad	complete metric spaces	Radon probability measures of finite first moment	Kantorovich-Wasserstein metric	closed convex subsets of Banach spaces	van Breugel '05, F-P '19
ordered Kantorovich monad	complete L-ordered? metric spaces	Radon probability measures of finite first moment	Kantorovich-Wasserstein metric, stochastic order	closed convex subsets of ordered Banach spaces?	F-P '20
Baire monad	Weakly Hausdorff quotients of countably-based topological spaces	Baire measures	topology induced by cartesian closedness	Full characterization unknown	K-P '24
Riesz monad	compactly generated weakly Hausdorff topological spaces	$k$-regular probability measures	topology induced by cartesian closedness	Full characterization unknown	K-P '24

(…to be expanded…)