The monad of probability measures over compact ordered spaces and its Eilenberg–Moore algebras

Abstract

The probability measures on compact Hausdorff spaces K form a compact convex subset P K of the space of measures with the vague topology. Every continuous map f : K → L of compact Hausdorff spaces induces a continuous affine map P f : P K → P L extending f. Together with the canonical embedding ε : K → P K associating to every point its Dirac measure and the barycentric map β associating to every probability measure on P K its barycenter, we obtain a monad ( P , ε , β ) . The Eilenberg–Moore algebras of this monad have been characterised to be the compact convex sets embeddable in locally convex topological vector spaces by Swirszcz [T. Swirszcz, Monadic functors and convexity, Bul. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 22 (1974) 39–42]. We generalise this result to compact ordered spaces in the sense of Nachbin [L. Nachbin, Topology and Order, Von Nostrand, Princeton, NJ, 1965. Translated from the 1950 monograph “Topologia e Ordem” (in Portugese). Reprinted by Robert E. Kreiger Publishing Co., Huntington, NY, 1967]. The probability measures form again a compact ordered space when endowed with the stochastic order. The maps ε and β are shown to preserve the stochastic orders. Thus, we obtain a monad over the category of compact ordered spaces and order preserving continuous maps. The algebras of this monad are shown to be the compact convex ordered sets embeddable in locally convex ordered topological vector spaces. This result can be seen as a step towards the characterisation of the algebras of the monad of probability measures on the category of stably compact spaces (see [G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, D.S. Scott, Continuous Lattices and Domains, Encyclopedia Math. Appl., vol. 93, Cambridge University Press, 2003, Section VI-6]).