Monad Structures on Topological Spaces Comprising Mislove's Random Variables

Mislove, Goubault and Varacca investigated how to define random variables in Domain theory to form monads over the category of bounded complete domains. They intended to model probabilistic programming languages with their random variables monads. In this paper, we focus on the random variables defined by Mislove from a topological perspective. We provide a topology for $\mathrm{\surd }$-max continuous random variables on a $T_0$ space, we construct a new $T_0$ space, where $\mathrm{\surd }$-max property is essential for the monad structures. We show that the spaces of normalized $\mathrm{\surd }$-max simple random variables form a monad over the category of $T_0$ spaces and that the spaces of normalized $\mathrm{\surd }$-max continuous random variables give a monad over the category of d-spaces. In addition, on a sober space, the space of normalized $\mathrm{\surd }$-max continuous random variables is the sobrification of the space of normalized $\mathrm{\surd }$-max simple random variables.