Monad Structures on Topological Spaces Comprising Mislove's Random Variables

Abstract: 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 $\surd$-max continuous random variables on a $T_0$ space, we construct a new $T_0$ space, where $\surd$-max property is essential for the monad structures. We show that the spaces of normalized $\surd$-max simple random variables form a monad over the category of $T_0$ spaces and that the spaces of normalized $\surd$-max continuous random variables give a monad over the category of d-spaces. In addition, on a sober space, the space of normalized $\surd$-max continuous random variables is the sobrification of the space of normalized $\surd$-max simple random variables.

• The paper constructs monads of normalized, surd-max simple random variables over T₀ spaces and continuous random variables over d-spaces, extending the theory to bounded complete domains.
• The surd-max restriction preserves continuity and order, while continuous random variables emerge as directed suprema of simple ones and as both the sobrification and D-completion of that space.
• The resulting monads are non-commutative, limiting direct modeling of independent parallel probabilistic choice and leaving adequacy, distributive laws, and cartesian closure as open problems.

Background and motivation

The paper addresses a long-standing problem in domain-theoretic semantics: constructing a monad of random variables suitable for modelling probabilistic computation. Mislove's earlier work introduced discrete random variables over domains [Mislove2005], and Goubault-Larrecq and Varacca proposed continuous random variables on bounded complete domains via thin valuations on the Cantor tree $\mathbb{C}$ [Goubault2011]; however, that construction fails to form a monad [Mislove2013, Mislove2014]. Mislove later repaired this by adjoining maximal finite elements marked with a termination symbol $\surd$ to obtain his Cantor tree $\mathbb{M} = \mathbb{C} \cup \mathbb{C}^{\surd}$ [Mislove2017]. Barker's randomized-choice monad [Barker2016] handles only non-determinism, while Di Gianantonio and Edalat's PER-domain construction [Pietro2024] solves the Jung–Tix problem by viewing random variables as push-forwards from fixed probability spaces. The present paper takes a different route: it interprets random variables as processes driven by an independent coin-flipping machine, adopts Mislove's Cantor tree as sample space (whose valuations cover all measures on Cantor space), and develops the construction over non-Hausdorff spaces — $T_0$ spaces, d-spaces, sober spaces, dcpos, and bounded complete domains.

Random variables with a topology

A continuous random variable on a $T_0$ space $X$ is a pair $(\mu, f)$ where $\mu \in \mathcal{V}_{\leq 1}\Sigma\mathbb{M}$ is a continuous valuation on Mislove's tree and $f$ is a partially continuous map with domain $supp(\mu)$. A lower subset $\surd$0 is called $\surd$1-max if every $\surd$2 has $\surd$3 maximal in $\surd$4; this condition excludes supports where a ticked string coexists with a strictly longer unticked extension. The space $\surd$5 of $\surd$6-max continuous random variables carries the topology generated by subbasic sets $\surd$7 requiring both $\surd$8 and $\surd$9. A key technical observation is that the topology generated by $\mathbb{M} = \mathbb{C} \cup \mathbb{C}^{\surd}$0 coincides with the weak (equivalently Scott) topology on $\mathbb{M} = \mathbb{C} \cup \mathbb{C}^{\surd}$1, which itself is shown to be a bounded complete domain whose basis elements enjoy binary meets. Consequently $\mathbb{M} = \mathbb{C} \cup \mathbb{C}^{\surd}$2 is $\mathbb{M} = \mathbb{C} \cup \mathbb{C}^{\surd}$3, its specialization order is the product order $\mathbb{M} = \mathbb{C} \cup \mathbb{C}^{\surd}$4 iff $\mathbb{M} = \mathbb{C} \cup \mathbb{C}^{\surd}$5 and $\mathbb{M} = \mathbb{C} \cup \mathbb{C}^{\surd}$6, and the valuation projection is continuous. Normalized variants $\mathbb{M} = \mathbb{C} \cup \mathbb{C}^{\surd}$7 and $\mathbb{M} = \mathbb{C} \cup \mathbb{C}^{\surd}$8 (simple random variables, i.e., those with valuations in the basis $\mathbb{M} = \mathbb{C} \cup \mathbb{C}^{\surd}$9) are defined analogously.

The simple-random-variable monad over $T_0$0 spaces

For a continuous map $T_0$1, the Kleisli lift $T_0$2 splices the distribution of $T_0$3 onto the subtree rooted at $T_0$4 for each ticked leaf $T_0$5, leaving mass on unticked nodes untouched. The unit is $T_0$6 with $T_0$7 mapping both $T_0$8 and $T_0$9 to $T_0$0. The authors verify all three Kleisli triple laws directly, establishing a monad $T_0$1 over $T_0$2.

Crucially, the $T_0$3-max restriction is not cosmetic. The paper exhibits an explicit counterexample showing that Mislove's original lift $T_0$4 on unrestricted simple random variables does not preserve the specialization order: for two particular two-point domains $T_0$5 and a specific $T_0$6, one obtains $T_0$7 yet $T_0$8, which would violate continuity of the lift. The $T_0$9-max property guarantees that the support of $X$0 decomposes as a disjoint union, making the case analysis in the definition of the lifted partial map coherent and order-preserving.

Sobrification, D-completion, and bounded complete domains

On the structural side, the paper proves that if $X$1 is sober then $X$2 is sober, using the characterization of sobriety via suprema of irreducible subsets; the argument relies on lemmas showing that directed suprema and irreducible suprema of $X$3-max closed subsets remain $X$4-max, and that $X$5 for irreducible families. If $X$6 is a d-space, then $X$7 and $X$8 are d-spaces.

The truncation maps $X$9 approximate any continuous random variable by simple ones: $(\mu, f)$0 on d-spaces. This yields two completion results:

• For sober $(\mu, f)$1, $(\mu, f)$2 is the sobrification of $(\mu, f)$3 (and $(\mu, f)$4 of $(\mu, f)$5), via Heckmann's criterion that every point of the larger space lies above a point of the dense subspace in each open set.
• For d-spaces $(\mu, f)$6, $(\mu, f)$7 is the D-completion of $(\mu, f)$8, since it is the minimal subdcpo containing the simple random variables.

Moreover, sobriety reflects: $(\mu, f)$9 is sober iff $\mu \in \mathcal{V}_{\leq 1}\Sigma\mathbb{M}$0 (equivalently $\mu \in \mathcal{V}_{\leq 1}\Sigma\mathbb{M}$1) is sober, giving a powerspace-style characterization of sober spaces. Over bounded complete domains $\mu \in \mathcal{V}_{\leq 1}\Sigma\mathbb{M}$2, $\mu \in \mathcal{V}_{\leq 1}\Sigma\mathbb{M}$3 is a bounded complete domain whose topology agrees with the Scott topology, and $\mu \in \mathcal{V}_{\leq 1}\Sigma\mathbb{M}$4 is a topological retract thereof (via normalization $\mu \in \mathcal{V}_{\leq 1}\Sigma\mathbb{M}$5), hence also a bounded complete domain with the Scott topology.

The continuous-random-variable monad over d-spaces

Extending the monad to continuous random variables requires more care because the naive lift need not land in $\mu \in \mathcal{V}_{\leq 1}\Sigma\mathbb{M}$6. The authors define, for $\mu \in \mathcal{V}_{\leq 1}\Sigma\mathbb{M}$7, the lift as a double directed supremum

$\mu \in \mathcal{V}_{\leq 1}\Sigma\mathbb{M}$8

which is well-defined and continuous on d-spaces thanks to the fact that composition in weak d-spaces preserves existing directed sups and that $\mu \in \mathcal{V}_{\leq 1}\Sigma\mathbb{M}$9 preserves bounded directed sups. With unit $f$0, this forms a Kleisli triple over the category $f$1 of d-spaces. An explicit formula for $f$2 is derived: mass on unticked nodes is retained, mass at each ticked leaf $f$3 is pushed through $f$4 applied to the valuation component of $f$5, and the map component is reassembled accordingly. Via the functors $f$6 and $f$7, this induces a Kleisli triple over $f$8 that restricts to $f$9 — a cartesian closed category — yielding what the authors regard as the principal contribution: a random-variable monad on bounded complete domains in which every continuous random variable is the supremum of a directed family of simple ones. This is presented as resolving Mislove's question from [Mislove2013].

Non-commutativity

Using Moggi's tensorial strengths, the paper computes the left and right strengths explicitly and shows they disagree: composing them produces $supp(\mu)$0 versus $supp(\mu)$1, which differ in general. Hence neither $supp(\mu)$2 nor $supp(\mu)$3 is commutative. This contrasts with the commutative monads of Di Gianantonio and Edalat [Pietro2024], and means these monads cannot directly model independent parallel probabilistic choice without additional structure.

Limitations and open questions

The paper concedes several gaps. No denotational semantics for a programming language is developed from either monad, so their computational adequacy remains untested. Whether a distributive law exists between these random-variable monads and the non-deterministic powerdomains is left open — a prerequisite for combining probability with demonic or angelic choice. Topologically, it is unknown whether $supp(\mu)$4 is core-compact whenever $supp(\mu)$5 is; the authors note this likely depends on the structure of spaces of partially continuous maps, and core-compactness would be needed for a cartesian closed category of such spaces. Finally, the class of spaces whose valuations arise entirely as push-forwards of continuous random variables is only partially characterized by Mislove's preliminary answer, and the paper conjectures the class is larger.

Conclusion

This paper transfers Mislove's random-variable construction from bounded complete domains to general non-Hausdorff spaces, equipping $supp(\mu)$6-max continuous random variables with a $supp(\mu)$7 topology generated by mixed valuation-and-value subbasic opens. It delivers monads of normalized $supp(\mu)$8-max simple random variables over $supp(\mu)$9 spaces and of normalized $\surd$00-max continuous random variables over d-spaces (hence over sober spaces, dcpos, and bounded complete domains), identifies the continuous-variable space simultaneously as sobrification and D-completion of the simple-variable space, and demonstrates via counterexamples both the necessity of the $\surd$01-max hypothesis and the failure of commutativity. The remaining semantic questions — adequacy, distributive laws, and cartesian closure — define the immediate agenda for this line of work.