# On Local Compactness of Spaces of Continuous Valuations

Jean Goubault-Larrecq

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## To cite this version:

Jean Goubault-Larrecq. On Local Compactness of Spaces of Continuous Valuations. Topology Proceedings, 2025, 65, pp.135-148. hal-04746726v2

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Submitted on 18 Nov 2024

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# ON LOCAL COMPACTNESS OF SPACES OF CONTINUOUS VALUATIONS

JEAN GOUBAULT-LARRECQ

**ABSTRACT.** We show that the spaces of continuous valuations, resp. subprobability valuations on a locally compact space is locally compact; similarly with probability valuations on locally compact, compact spaces. Continuous valuations are close cousins of measures. No separation property is assumed.

## 1. Introduction

A continuous valuation on a topological space $X$ is a close cousin of a Borel measure. Continuous valuations on $X$ form a space $VX$, with a topology known as the weak topology, and similarly for the subspaces $V_{\le 1}X$ of subprobability valuations and $V_1X$ of probability valuations. We will define these notions more precisely below. It is known that $V_{\le 1}$ preserves various properties: stable compactness [2, Theorem 39], being a continuous dcpo [16, Theorem 5.2], being a quasi-continuous dcpo [14, Theorem 5.1], for example. Some of these preservation theorems extend over to $V_1$ or to $V$, but a conspicuously absent property in the list is local compactness. This is what we address in this paper.

The proof relies on credibilities, as already used in [14], on the Smyth powerdomain construction [23], on Jones' celebrated theorem that the probabilistic powerdomain preserves continuous dcpos [16, Theorem 5.2], and on a few things about certain functionals called superlinear previsions [12]. As a whole, the proof is therefore rather elaborate; but it also results from little more than an assemblage of previously known results, and as a consequence, is pretty short.

## 2. Preliminaries

For background on topology, we refer the reader to [11].

A *compact* subset $A$ of a space $X$ is one such that one can extract a finite subcover from any of its open covers. No separation property is

<footnote n="2010">2010 Mathematics Subject Classification. Primary 54G99; secondary 28A33, 68Q55.</footnote>

<footnote>*Key words and phrases.* Continuous valuations, spaces of measures, local compactness, previsions.</footnote>
assumed. A subset $A$ of $X$ is *saturated* if and only if it is equal to the intersection of its open neighborhoods, or equivalently if and only if it is upwards-closed in the specialization preordering of $X$. The latter is defined by $x \le y$ if and only if every open neighborhood of $x$ contains $y$.

A space $X$ is *locally compact* if and only if every point has a base of compact neighborhoods, or equivalently of compact saturated neighborhoods, since for any compact subset $K$ of $X$, the upward closure $\uparrow K$ of $K$ with respect to the specialization preordering of $X$ is compact saturated. In non-Hausdorff spaces, a compact space may fail to be locally compact.

A space is $T_0$ if and only if its specialization preordering is antisymmetric. An *irreducible* closed subset $C$ of $X$ is a non-empty closed subset such that, for any two closed subsets $C_1$ and $C_2$ of $X$ such that $C \subseteq C_1 \cup C_2$, $C$ is included in $C_1$ or in $C_2$ already; equivalently, if $C$ intersects two open sets, it must intersect their intersection. A space $X$ is *sober* if and only if it is $T_0$ and every irreducible closed subset is of the form $\downarrow x$ for some point $x \in X$. Every Hausdorff space, for example, is sober. The notation $\downarrow x$ stands for the *downward closure* of $x$ in $X$, namely the set of points $y$ below $x$; that is also the closure of $\{x\}$. Symmetrically, $\uparrow x$ stands for the *upward closure* of $x$, namely the set of points $y$ above $x$. This notation extends to $\uparrow A$, for any subset $A$, denoting $\bigcup_{x \in A} \uparrow x$.

A function $f: P \to Q$ between posets is *monotonic* if and only if for all $x, x' \in P$, $x \le x'$ implies $f(x) \le f(x')$. It is *Scott-continuous* if and only if $f$ is monotonic and for every directed family $(x_i)_{i \in I}$ with a supremum $x$ in $P$, the (necessarily directed) family of elements $f(x_i)$ has $f(x)$ as supremum. A *directed* family is a non-empty family $D$ such that any two elements of $D$ have an upper bound in $D$. We write $\sup_D$ for the supremum of a directed family $D$. Scott-continuity is equivalent to continuity with the respective Scott topologies on $P$ and $Q$. The *Scott topology* on a poset $P$ consists of those subsets $U$ that are upwards closed ($x \in U$ and $x \le x'$ implies $x' \in U$) and such that every directed family $D$ that has a supremum in $U$ intersects $U$.

A dcpo (short for *directed-complete partial order*) is a poset $P$ in which every directed subset has a supremum. In a dcpo $P$, let $x \ll y$ (“$x$ is *way below* $y$”) if and only if every directed family $D$ such that $y \le \sup_D$ contains an element $d \in D$ such that $x \le d$. A *basis* for $P$ is a subset $B$ such that, for every $x \in P$, $\mathfrak{d}Bx \stackrel{\text{def}}{=} \{b \in B \mid b \ll x\}$ is directed and has $x$ as its supremum. A dcpo is *continuous* if and only
if it has a basis. If so, the sets $\uparrow b \stackrel{\text{def}}{=} \{x \in P \mid b \ll x\}$ form a base of the Scott topology.

Let $\overline{\mathbb{R}}_+$ be the set of extended non-negative real numbers $\mathbb{R}_+ \cup \{\infty\}$, with its usual ordering. When needed, we will consider it with its Scott topology, whose open sets are the intervals $[t, \infty]$, $t \in \mathbb{R}_+$, plus $\emptyset$ and $\overline{\mathbb{R}}_+$ itself. In passing, $\overline{\mathbb{R}}_+$ is an example of a continuous dcpo, with $s \ll t$ if and only if $s = 0$ or $s < t$.

We write $\mathcal{O}X$ for the lattice of open subsets of a space $X$. A continuous valuation on $X$ is a map $\nu: \mathcal{O}X \to \overline{\mathbb{R}}_+$ that is strict ($\nu(\emptyset) = 0$), modular (for all $U, V \in \mathcal{O}X$, $\nu(U) + \nu(V) = \nu(U \cup V) + \nu(U \cap V)$) and Scott-continuous. We say that $\nu$ is bounded if $\nu(X) < \infty$, a probability valuation if and only if $\nu(X) = 1$, and a subprobability valuation if and only if $\nu(X) \le 1$.

Continuous valuations are an alternative to measures that have become popular in domain theory [17, 16]. The first results that connected continuous valuations and measures are due to Saheb-Djahromi [21] and Lawson [19]. The current state of the art on this matter is the following. In one direction, every measure on the Borel $\sigma$-algebra of $X$ induces a continuous valuation on $X$ by restriction to the open sets, if $X$ is hereditarily Lindelöf (namely, if every directed family of open sets contains a cofinal monotone sequence). This is an easy observation, and one half of Adamski's theorem [1, Theorem 3.1], which states that a space is hereditary Lindelöf if and only if every measure on its Borel $\sigma$-algebra restricts to a continuous valuation on its open sets. In the other direction, every continuous valuation on a space $X$ extends to a measure on the Borel sets provided that $X$ is an LCS-complete space [6, Theorem 1]. An LCS-complete space is a $G_\delta$ subspace of a locally compact sober space, and the class of LCS-complete spaces contains all locally compact sober spaces, in particular all continuous dcpos from domain theory, all of M. de Brecht's quasi-Polish spaces [13] and therefore all Polish spaces.

The extension of a continuous valuation $\nu$ to a measure on the Borel $\sigma$-algebra is unique if $\nu$ is bounded. This is a consequence of the $\pi$-$\lambda$ theorem [3, Theorem 3.2], usually applied to probability measures [3, Theorem 3.3], but the same proof works for bounded measures.

Hence the above maps from Borel measures to continuous valuations and conversely, defined on hereditary Lindelöf LCS-complete spaces (in particular, quasi-Polish spaces), are inverse of each other when restricted to bounded measures and bounded continuous valuations.

Let $VX$ denote the space of continuous valuations on a space $X$, with the *weak topology*, defined by subbasic open sets $[U > r] \stackrel{\text{def}}{=} \{\nu \in$
$VX$ | $\nu(U) > r\}$, where $U \in \mathcal{O}X$ and $r \in \mathbb{R}_+$. We define its subspace $V_1X$ of probability valuations and $V_{\le 1}X$ (subprobability) similarly. In general, we write $V_{\bullet}X$, where $\bullet$ can be nothing, "$\le 1$", or "1". The specialization ordering of each is the stochastic ordering $\le$ given by $\nu \le \nu'$ if and only if $\nu(U) \le \nu'(U)$ for every $U \in \mathcal{O}X$; indeed, $\nu \le \nu'$ if and only if for every $U \in \mathcal{O}X$, for every $r \in \mathbb{R}_+$, $\nu \in [U > r]$ implies $\nu' \in [U > r]$.

There is a related topology on spaces of probability measures. A sequence of probability measures $P_n, n \in \mathbb{N}$, on a metric space converges weakly to a probability measure $P$ if and only if $\liminf_{n \in \mathbb{N}} P_n(U) \ge P(U)$ for every open subset $U$ [4, Portmanteau Theorem 2.1]. It is easy to see that this is equivalent to: for every open subset $U$ and for every $r \in \mathbb{R}_+$, $P(U) > r$ implies that $P_n(U) > r$ for $n$ large enough. Hence we may define the weak topology on spaces of measures (not just probability measures) on a topological space $X$ (not just a metric space) as given via a subbasis of sets $\{P \mid P(U) > r\}, U \in \mathcal{O}X, r \in \mathbb{R}_+$. We obtain the following.

**Proposition 2.1.** The weak topology makes the spaces $M_{\le 1}X$ and $M_1X$ of subprobability, resp. probability measures homeomorphic to $V_{\le 1}X$ and $V_1X$, respectively, for every hereditarily Lindelöf LCS-complete space $X$.

$V_{\bullet}$ is the object part of an endofunctor on the category of topological spaces, whose morphism part is defined as follows: for every continuous map $f: X \to Y$, $V_{\bullet}f$ maps every $\nu \in V_{\bullet}X$ to its image valuation $f[\nu]$, where $f[\nu](V) \stackrel{\text{def}}{=} \nu(f^{-1}(V))$ for every $V \in \mathcal{O}Y$.

We will prove slightly more than the fact that $V_{\bullet}$ preserves local compactness, where $\bullet$ is nothing, "$\le 1$", or "1". A space $X$ is core-compact if and only if $\mathcal{O}X$ is a continuous dcpo; every locally compact space is core-compact [11, Theorem 5.2.9]. The connection between the two notions can be made more precise as follows. Every topological space $X$ has a sobrification $S(X)$ (or $X^s$), which is the free sober space over $X$ [11, Theorem 8.2.44]; then $X$ is core-compact if and only if $S(X)$ is locally compact [11, Proposition 8.3.11]. $S(X)$ can be built as the collection of irreducible closed subsets, with the topology whose open sets (all of them, not just a base) are $\diamond U \stackrel{\text{def}}{=} \{F \in S(X) \mid F \cap U \ne \emptyset\}$, $U \in \mathcal{O}X$. In particular, $\diamond: U \mapsto \diamond U$ is an order-isomorphism between $\mathcal{O}X$ and $\mathcal{OS}(X)$. This induces a homeomorphism between $V_{\bullet}X$ and $V_{\bullet}\mathcal{S}(X)$.
### 3. Credibilities, previsions and the Smyth powerdomain

The proof of our main theorem will rely on reducing the question to spaces of valuations on Smyth powerdomains, and on using notions of credibilities [10] and of previsions [9, 12]. We need to give a quick tour of the part of that theory that we need.

Let $\mathcal{L}X$ denote the dcpo of all lower semicontinuous maps from $X$ to $\overline{\mathbb{R}}_+; $ a map $f: X \to \overline{\mathbb{R}}_+$ is lower semicontinuous if and only if it is continuous, where $\overline{\mathbb{R}}_+$ is given its Scott topology. The ordering on $\mathcal{L}X$ is the pointwise ordering. For every monotone map $\nu: \mathcal{O}X \to \overline{\mathbb{R}}_+$, for every $h \in \mathcal{L}X$, there is a Choquet integral $\int_{x \in X} h(x) \, d\nu$, defined as the indefinite Riemann integral $\int_0^\infty \nu(h^{-1}([t, \infty])) \, dt$. We need to reprove in part some of the results of [10] on Choquet integrals, since that paper only considers integrals of maps from $X$ to $\overline{\mathbb{R}}_+; $ we will do this in Lemma 3.1 below.

A *prevision* on a space *X* is a Scott-continuous map *F*: *L*X → *R*<sub>+</sub> that is *positively homogeneous* in the sense that *F*(ah) = *aF*(h) for all *a* ∈ *R*<sub>+</sub> and *h* ∈ *L*X. There is a space *PX* of previsions on *X*, whose topology is generated by sets [*h* > *r*] def= {*F* | *F*(h) > *r*}, *h* ∈ *L*X, *r* ∈ *R*<sub>+</sub>.

For example, any continuous valuation $\nu$ on $X$ gives rise to a prevision $G: h \mapsto \int_{x \in X} h(x) \, d\nu$. Such a prevision is *linear*, in the sense that $G(h + h') = G(h) + G(h')$ for all $h, h' \in \mathcal{L}X$. Let $\mathbb{P}_X$ be the subspace of $\mathbb{P}X$ of linear previsions. In the reverse direction, every linear prevision $G \in \mathbb{P}_X$ gives rise to a continuous valuation $U \mapsto G(\chi_U)$, where $\chi_U$ is the characteristic map of the open set $U$, and the two constructions are inverse of each other. Additionally, those two constructions define continuous maps between $\mathbf{V}X$ and $\mathbb{P}_X$ [24, Satz 4.16]. We will therefore equate continuous valuations with linear previsions.

A prevision is *superlinear* if and only if $G(h + h') \geq G(h) + G(h')$ for all $h, h' \in \mathcal{L}X$. As in [12], we write $\mathbb{P}_{\mathrm{DP}}X$ for the subspace of $\mathbb{P}X$ consisting of superlinear previsions.

Among the continuous valuations, there are the probability valuations and the subprobability valuations. Similarly, we say that a prevision $F$ is *subnormalized* (resp., *normalized*) iff $F(\mathbf{1} + h) \leq \mathbf{1} + F(h)$ (resp., $=$) for every $h \in \mathcal{L}X$, where $\mathbf{1}$ is the constant function with value 1. The homeomorphism between $\mathbf{V}X$ and $\mathbb{P}_X$ restricts to homeomorphisms between $\mathbf{V}_{\leq 1}X$ (resp., $\mathbf{V}_1X$) and the subspace $\mathbb{P}_\mathbf{P}^{\leq 1}X$ (resp., $\mathbb{P}_\mathbf{P}^1X$) of subnormalized (resp., normalized) linear previsions on $X$. We
write $\mathbb{P}_{\mathrm{DP}}^{\le 1} X$, $\mathbb{P}_{\mathrm{DP}}^{1} X$ for the corresponding spaces of (sub)normalized, superlinear previsions. In general, we write $\mathbb{P}_{\mathrm{DP}}^{\bullet} X$, where $\bullet$ can be nothing, "$\le 1$", or "$1$".

For every topological space $X$, let $\mathcal{Q}X$ be the collection of non-empty compact saturated subsets of $X$, ordered by reverse inclusion $\supseteq$. This is the *Smyth powerdomain* of $X$ [23]; see also [22, Section 7]. Let $\mathcal{Q}\mathcal{V}X$ be the same set, but with the *upper Vietoris topology*, whose basic open subsets are $\Box U \stackrel{\mathrm{def}}{=} \{Q \in \mathcal{Q}X \mid Q \subseteq U\}$, $U \in \mathcal{O}X$. We will first use this in item (6) of Lemma 3.1 below. The functions $\sum_{j=1}^{m} a_j u_{Q_j}$ of item (7) of the same lemma are the (simple) *credibilities* of [10, Theorem 6.18].

**Lemma 3.1.** The following properties hold, where $h \in \mathcal{L}X$ and $\nu$ is a monotonic map from $\mathcal{O}X$ to $\bar{\mathbb{R}}_+$.

(1) The Choquet integral $\int_{x \in X} h(x) \, d\nu$ is linear in $\nu$, monotonic and even Scott-continuous in $\nu$.

(2) If $\nu$ is Scott-continuous, then the Choquet integral is Scott-continuous in $h$.

(3) If $\nu$ is a continuous valuation, then the Choquet integral is also linear in $h$.

(4) For every $U \in \mathcal{O}X$, $\int_{x \in X} \chi_U(x) \, d\nu = \nu(U)$.

(5) Given any continuous valuation $\mu$ on $\mathcal{Q}\mathcal{V}X$, there is a Scott-continuous map $\mu^{\square}: \mathcal{O}X \to \bar{\mathbb{R}}_+$ defined by $\mu^{\square}(U) \stackrel{\mathrm{def}}{=} \mu(\square U)$ for every $U \in \mathcal{O}X$. Then, for every $h \in \mathcal{L}X$, the map $h^*: Q \mapsto \min_{x \in Q} h(x)$ is in $\mathcal{L}\mathcal{Q}\mathcal{V}X$ and $\int_{x \in X} h(x) \, d\mu^{\square} = \int_{Q \in \mathcal{Q}\mathcal{V}X} h^*(Q) \, d\mu$.

(6) For every compact saturated subset $Q$ of $X$, the unanimity game $u_Q: \mathcal{O}X \to \bar{\mathbb{R}}_+$, which maps every $U \in \mathcal{O}X$ to 1 if $Q \subseteq U$ and to 0 otherwise, is a Scott-continuous map from $\mathcal{O}X$ to $\bar{\mathbb{R}}_+$.

(7) Letting $\nu \stackrel{\mathrm{def}}{=} \sum_{j=1}^{m} a_j u_{Q_j}$, where each $Q_j$ is compact saturated and $a_j \in \bar{\mathbb{R}}_+$, the map $F: \mathcal{L}X \to \bar{\mathbb{R}}_+$ defined by:

$$F(h) \stackrel{\mathrm{def}}{=} \int_{x \in X} h(x) \, d\nu = \sum_{j=1}^{m} a_j \min_{x \in Q_j} h(x)$$

for every $h \in \mathcal{L}X$, is a superlinear prevision.

*Proof.* (1) The fact that the Choquet integral is linear in $\nu$, namely that it commutes with scalar products by non-negative real numbers and with addition of continuous valuations, follows from the linearity of indefinite Riemann integration. It is also monotonic in $\nu$. In order to show Scott-continuity, we consider a directed family $(\nu_i)_{i \in I}$, with (pointwise) supremum $\nu$, and we observe that $\int_{x \in X} h(x) \, d\nu = \int_{0}^{\infty} \sup_{i \in I} \nu_i(h^{-1}([t, \infty])) \, dt$. The key is that the maps $t \mapsto \nu_i(h^{-1}([t, \infty]))$
are antitone (all antitone maps are Riemann-integrable), and that indefinite Riemann integration of antitone maps $f$ is Scott-continuous in $f$, see [24, Lemma 4.2]. Therefore $\int_{x \in X} h(x) d\nu = \sup_{i \in I} \int_0^\infty \nu_i(h^{-1}([t, \infty])) dt = \sup_{i \in I} \int_{x \in X} h(x) d\nu_i$.

(2) The proof works as Tix's original proof of the same result in the special case where $\nu$ is a continuous valuation [24, Satz 4.4], and relies on [24, Lemma 4.2], just like item (1). Explicitly, let $(h_i)_{i \in I}$ be a directed family in $\mathcal{L}X$, with (pointwise) supremum $h$. For every $t \in \mathbb{R}_+$, $h^{-1}([t, \infty)) = \{x \in X \mid \sup_{i \in I} h_i(x) > t\} = \bigcup_{i \in I} h_i^{-1}([t, \infty])$. Therefore,

$$ 
\begin{align*}
\int_{x \in X} h(x) d\nu &= \int_0^\infty \nu(\bigcup_{i \in I} h_i^{-1}([t, \infty])) dt \\
&= \int_0^\infty \sup_{i \in I} \nu(h_i^{-1}([t, \infty])) dt \\
&= \sup_{i \in I} \int_0^\infty \nu(h_i^{-1}([t, \infty])) dt = \sup_{i \in I} \int_{x \in X} h_i(x) d\nu,
\end{align*}
$$

using the Scott-continuity of $\nu$ and the Scott-continuity of indefinite Riemann integration of antitone maps.

(3) This is a result of Tix [24, Satz 4.4].

(4) $\int_{x \in X} \chi_U(x) d\nu = \int_0^\infty \nu(\chi_U^{-1}([t, \infty])) dt = \int_0^1 \nu(U) dt + \int_1^\infty 0 dt = \nu(U)$.

(5) This is as with [10, Lemma 7.5]. The fact that $\mu^\square$ is Scott-continuous follows from the fact that $\mu$ is, and that the $\square$ operator is, too. For the latter, observe that for every directed family $(U_i)_{i \in I}$ of open subsets of $X$, for every $Q \in \mathcal{Q}\sqrt{X}$, $Q \in \square \bigcup_{i \in I}^\uparrow U_i$ if and only if $Q \subseteq \bigcup_{i \in I}^\uparrow U_i$, which is equivalent to $Q \subseteq U_i$ (namely, $Q \in \square U_i$) for some $i \in I$, because $Q$ is compact.

For every $Q \in \mathcal{Q}\sqrt{X}$, the minimum of $h(x)$ when $x$ ranges over $Q$ is reached, since $Q$ is compact and non-empty. For every $t \in \mathbb{R}_+$, $Q \in h^{*-1}([t, \infty))$ if and only if $h^*(Q) > t$. The latter certainly implies that $h(x) > t$ for every $x \in Q$, hence that $Q \in \square h^{-1}([t, \infty])$. Conversely, if $Q \in \square h^{-1}([t, \infty])$, then let us pick $x \in Q$ such that $h(x)$ is the least value reached by $h$ on $Q$; then $h^*(Q) = h(x) > t$, so $Q \in h^{*-1}([t, \infty])$. Hence $h^{*-1}([t, \infty]) = \square h^{-1}([t, \infty])$. This implies that $h^*$ is in $\mathcal{L}\mathcal{Q}\sqrt{X}$, in particular.
Now:

$$
\begin{align*}
\int_{Q \in \mathcal{Q}_V X} h^*(Q) \, d\mu &= \int_0^\infty \mu(h^{*-1}([t, \infty])) \, dt \\
&= \int_0^\infty \mu(\Box h^{-1}([t, \infty])) \, dt \\
&= \int_0^\infty \mu^\Box(h^{-1}([t, \infty])) \, dt = \int_{x \in X} h(x) \, d\mu^\Box.
\end{align*}
$$

(6) Monotonicity is clear. For every directed family $(U_i)_{i \in I}$ of open subsets of $X$, $u_Q(\bigcup_{i \in I} U_i) = 1$ if and only if $Q \in \Box \bigcup_{i \in I} U_i$, which is equivalent to the existence of an $i \in I$ such that $Q \in \Box U_i$ (equivalently, $u_Q(U_i) = 1$), as we have seen at the beginning of the proof of item (5).

(7) That would be a consequence of [10, Propositions 7.2, 7.6], except for the fact that our functions $h$ take their values in $\bar{\mathbb{R}}_+$. We verify that $\int_{x \in X} h(x) \, d\nu = \sum_{j=1}^m a_j \int_{x \in X} h(x) \, du_{Q_j} = \sum_{j=1}^m a_j \min_{x \in Q_j} h(x)$: the first equality is by item (1), and the second one is because $\int_{x \in X} h(x) \, du_{Q_i}$ is equal to $\int_0^\infty u_{Q_i}(h^{-1}([t, \infty])) \, dt = \int_0^{\min_{x \in Q_i} h(x)} 1 \, dt + \int_{\min_{x \in Q_i} h(x)}^\infty 0 \, dt = \min_{x \in Q_i} h(x)$.

It is easy to see that $F(h)$ is superlinear, because $\min_{x \in Q_i} ah(x) = a \min_{x \in Q_i} h(x)$ and $\min_{x \in Q_i}(h(x)+h'(x)) \ge \min_{x \in Q_i} h(x)+\min_{x \in Q_i} h'(x)$, for all $a \in \mathbb{R}_+$ and $h,h' \in \mathcal{L}X$. Scott-continuity comes from the fact that $F(h) = \int_{x \in X} h(x) \, d\nu$, where $\nu = \sum_{j=1}^m a_j u_{Q_j}$, that $\nu$ is Scott-continuous (by item (6)), and by using item (2). $\square$

Let us turn to $\mathcal{Q}_V$. The specialization preordering of $\mathcal{Q}_V X$ is $\supseteq$, and in particular $\mathcal{Q}_V X$ is $T_0$. The Scott topology on $\mathcal{Q}X$ is finer than the upper Vietoris topology when $X$ is sober, and coincides with it when $X$ is locally compact and sober [11, Lemma 8.3.26], and then $\mathcal{Q}X$ itself is a continuous dcpo [11, Proposition 8.3.25].

$\mathcal{Q}_V$ is an endofunctor on the category of topological spaces, whose action on continuous maps $f: X \to Y$ is given by $\mathcal{Q}_V f(Q) \stackrel{\text{def}}{=} \uparrow f[Q]$, where $f[Q]$ denotes the image of $Q$ under $f$. That functor is part of a monad [22, Section 7], of which we need to know the unit $\eta^Q$: for every space $X$, for every $x \in X$, $\eta_X^Q(x) = \uparrow x$. We also note that $(\eta_X^Q)^{-1}(\Box U) = U$ for every $U \in \mathcal{O}X$.

By [12, Lemma 3.20], for every space $X$, there is a continuous map $s_{\bullet, \mathbb{P}}: \mathbb{P}_{\bullet, \mathbb{P}}^\bullet X \to \mathcal{Q}_V \mathbb{P}_{\bullet, \mathbb{P}}^\bullet X$ defined by $s_{\bullet, \mathbb{P}}(F) \stackrel{\text{def}}{=} \{G \in \mathbb{P}_{\bullet, \mathbb{P}}^\bullet X \mid G \ge F\}$. The ordering $\le$ between previsions is the specialization ordering, which is pointwise, and $\ge$ is the opposite ordering. Up to the homeomorphism between $\mathbb{P}_{\bullet, \mathbb{P}}^\bullet X$ and $\mathbf{V}_{\bullet, \mathbb{X}}$, we restate the latter as follows.
**Lemma 3.2.** For every topological space X, there is a continuous map $s_{\mathrm{DP}}^{\bullet}: \mathbb{P}_{\mathrm{DP}}^{\bullet}X \to Q_{\vee}V_{\bullet}X$, defined by:

$$s_{\mathrm{DP}}^{\bullet}(F) \stackrel{\mathrm{def}}{=} \left\{ \nu' \in V_{\bullet}X \mid \forall h \in \mathcal{L}X, \int_{x \in X} h(x) \, d\nu' \geq F(h) \right\}.$$

In order to show that $V_{\bullet}X$ is locally compact (if X is), we will need to find enough compact subsets. They will be provided to us by the images of certain superlinear previsions under $s_{\mathrm{DP}}^{\bullet}$.

## 4. The main theorem

We spend most of this section proving the following theorem. We remember that all locally compact spaces are core-compact.

**Theorem 4.1.** For every core-compact space X, $VX$ and $V_{\leq 1}X$ are locally compact. If X is also compact, then $V_1X$ is locally compact.

*Proof.* Replacing X by $S(X)$ if necessary, we may assume that X is locally compact and sober.

A fundamental theorem due to Jones [16, Theorem 5.2] states that for every continuous dcpo P, $V_{\leq 1}P$ is a continuous dcpo under the stochastic ordering, and that a basis is given by the simple valuations, namely those of the form $\sum_{i=1}^n a_i \delta_{x_i}$, where each $a_i$ is in $\mathbb{R}_+$ and $x_i \in P$. The notation $\delta_x$ refers to the Dirac valuation at $x$, defined by $\delta_x(U) \stackrel{\mathrm{def}}{=} 1$ if $x \in U$, 0 otherwise, for every $U \in \mathcal{O}X$.

A similar result holds for $VP$ [8, Theorem IV-9.16], and for $V_1P$ provided that P is also pointed [7, Corollary 3.3]. (A poset is pointed if and only if it has a least element.) We will apply those results to $P \stackrel{\mathrm{def}}{=} QX$, and we notice that if X is compact, then P is pointed, as X itself will be the least element of P in that case.

Let $\nu \in V_{\bullet}X$, and let $\mathcal{U}$ be any open neighborhood of $\nu$. Then $\nu$ is in some finite intersection of subbasic open sets $\bigcap_{i=1}^n [U_i > r_i]$ that is included in $\mathcal{U}$, where each $U_i$ is open in X and $r_i \in \mathbb{R}_+$. We consider $\mu \stackrel{\mathrm{def}}{=} V\eta_X^Q(\nu) \in V_{\bullet}Q_X X$. We recall that $\eta_X^Q$ is the unit of the $Q_X$ monad, and maps every point $x \in X$ to $\uparrow x \in Q_X X$. For every open subset U of X, $\mu(\Box U) = \eta_X^Q[\nu](\Box U) = \nu((\eta_X^Q)^{-1}(\Box U)) = \nu(U)$. It follows that $\mu$ is in $\bigcap_{i=1}^n [\Box U_i > r_i]$. The latter is open in the upper Vietoris topology on $V_{\bullet}Q_X X = V_{\bullet}P$, hence in the Scott topology of the stochastic ordering. Since X is locally compact and sober, P is a continuous dcpo, so $V_{\bullet}P$ is a continuous dcpo with a basis of simple valuations, and therefore there is a simple valuation $\xi \stackrel{\mathrm{def}}{=} \sum_{j=1}^m a_j \delta_{Q_j}$ in $V_{\bullet}P$ that is way below $\mu$ and in $\bigcap_{i=1}^n [\Box U_i > r_i]$.
We may form $\xi^{\square}$, as in Lemma 3.1, item (5): for every $U \in \mathcal{O}X$,
$\xi^{\square}(U) = \xi(\square U)$. We check easily that $\xi^{\square}$ is equal to $\sum_{j=1}^{m} a_j u_{Q_j}$. Using
Lemma 3.1, item (7) allows us to build a superlinear prevision $F$ on
$X$ by letting $F(h) \stackrel{\text{def}}{=} \int_{x \in X} h(x) \, d\xi^{\square} = \sum_{j=1}^{m} a_j \min_{x \in Q_j} h(x)$ for every
$h \in \mathcal{L}X$.

Now $s_{\mathrm{DP}}^{\bullet}(F)$ is a compact saturated subset of $\mathbf{V}_{\bullet}X$, by Lemma 3.2.
We claim that $\nu$ is in the interior of $s_{\mathrm{DP}}^{\bullet}(F)$, and that $s_{\mathrm{DP}}^{\bullet}(F)$ is included
in $\mathcal{U}$; this will end our proof.

We start by showing that $s_{\mathrm{DP}}^{\bullet}(F) \subseteq \mathcal{U}$. Let $\nu'$ be any element of
$s_{\mathrm{DP}}^{\bullet}(F)$. In other words, for every $h \in \mathcal{L}X$, $\int_{x \in X} h(x) \, dv' \ge F(h) =
\sum_{j=1}^{m} a_j \min_{x \in Q_j} h(x)$. For each $i \in \{1, \dots, n\}$, we apply this to $h \stackrel{\text{def}}{=}
\chi_{U_i}$; then $\min_{x \in Q_j} h(x) = u_{Q_j}(U_i)$, so $F(\chi_{U_i}) = \sum_{j=1}^{m} a_j u_{Q_j}(U_i) =
\xi^{\square}(U_i) = \xi(\square U_i)$, and therefore $\nu'(U_i) = \int_{x \in X} \chi_{U_i}(x) \, dv' \ge F(\chi_{U_i}) =
\xi(\square U_i) > r_i$, since $\xi \in [\square U_i > r_i]$. Hence $\nu' \in \bigcap_{i=1}^{n} [U_i > r_i] \subseteq \mathcal{U}$.

Next, we verify that $\nu$ is in the interior of $s_{\mathrm{DP}}^{\bullet}(F)$. We use the fact that
$\xi$ is way below $\mu$, equivalently that $\mu$ is in the open set $\upharpoonright\xi$. Since $\mu =
\mathbf{V}\eta_X^{\mathcal{Q}}(\nu)$, $\nu$ is in $(\mathbf{V}\eta_X^{\mathcal{Q}})^{-1}(\upharpoonright\xi)$, which is open since $\mathbf{V}\eta_X^{\mathcal{Q}}$ is continuous.
It remains to show that $(\mathbf{V}\eta_X^{\mathcal{Q}})^{-1}(\upharpoonright\xi)$ is included in $s_{\mathrm{DP}}^{\bullet}(F)$.

For every $\nu' \in (\mathbf{V}\eta_X^{\mathcal{Q}})^{-1}(\upharpoonright\xi)$, by definition $\xi$ is way below, in particular less than or equal to $\mu' \stackrel{\text{def}}{=} \mathbf{V}\eta_X^{\mathcal{Q}}(\nu')$. The latter is such that
$\mu'(\square U) = \eta_X^{\mathcal{Q}}[\nu'](\square U) = \nu'((\eta_X^{\mathcal{Q}})^{-1}(\square U)) = \nu'(U)$ for every $U \in \mathcal{O}X$. Hence $\nu' = \mu'^{\square}$. By Lemma 3.1, item (5), $\int_{x \in X} h(x) \, dv' = \int_{Q \in \mathcal{Q} \lor X} h^*(Q) \, d\mu'$. Since $\xi \le \mu'$, the latter is larger than or equal to $\int_{Q \in \mathcal{Q} \lor X} h^*(Q) \, d\xi = \int_{x \in X} h(x) \, d\xi^{\square} = F(h)$, using Lemma 3.1, item (5) once again. We have shown that $\int_{x \in X} h(x) \, dv' \ge F(h)$ for every $h \in \mathcal{L}X$, so $\nu' \in s_{\mathrm{DP}}^{\bullet}(F)$, as promised. $\square$

When $X$ is compact, $\mathbf{V}_1X = s_{\mathrm{DP}}^{\bullet}(F)$, where $F$ is the normalized
superlinear prevision $h \in \mathcal{L}X \mapsto \int_{x \in X} h(x) \, du_X = \min_{y \in X} h(y)$. In-
deed, for every probability valuation $\nu$ on $X$, for every $h \in \mathcal{L}X$,
$\int_{x \in X} h(x) \, dv \ge \int_{x \in X} \min_{y \in X} h(y) \, dv = \min_{y \in X} h(y)$. We therefore ob-
tain the following.

**Fact 4.2.** If $X$ is compact, then $\mathbf{V}_1X$ is compact.

**Lemma 4.3.** Let $\bullet$ be nothing, "$\le 1$", or "1". For every space $X$, $\mathbf{V}_{\bullet}X$ is sober.

*Proof.* For every space $X$, $\mathbf{V}X$ is sober. The argument is due to R. Tix [24, Satz 5.4], following ideas by R. Heckmann (see [15, Section 2.3]). The sober subspaces of a sober space $Z$ coincide with the subsets that are closed in the strong topology on $Z$ [18, Corollary 3.5]. The latter
is also known as the Skula topology, and is the smallest one generated
by the original topology on Z and all the downwards-closed subsets. In
particular, any closed subspace of a sober space is sober, any saturated
subspace of a sober space is sober. Hence $V_{\le 1}Y$ is sober, being equal
to the closed subspace $VY \setminus [Y > 1]$ of $VY$, and $V_1Y$ is sober, being
upwards-closed in $V_{\le 1}Y$. $\square$

**Corollary 4.4.** *V and $V_{\le 1}$ restrict to endofunctors on the category of locally compact (resp., locally compact sober) spaces and continuous maps. $V_1$ restricts to an endofunctor on the category of compact, locally compact (resp., compact, locally compact sober) spaces.*

We finish with the mandatory application to spaces of measures,
which follows directly, using the considerations developed in Section 2
on the relationship between measures and continuous valuations.

**Corollary 4.5.** For every hereditarily Lindelöf, LCS-complete space $X$—in particular, for every quasi-Polish space $X$—if $X$ is locally compact, then $M_{\le 1}X \cong V_{\le 1}X$ are locally compact; if $X$ is locally compact and compact, then $M_1X \cong V_1X$ are locally compact and compact.

The locally compact, quasi-Polish spaces are exactly the locally com-
pact, sober, second-countable spaces [5, Theorem 44]. It is easy to see
that the $V_\bullet$ functor preserves second countability. Using Lemma 4.3,
we obtain our final corollary.

**Corollary 4.6.** For every locally compact quasi-Polish space $X$, $M_{\le 1}X \cong V_{\le 1}X$ are locally compact and quasi-Polish.

5. CONCLUSION

When trying to show that a space Z is locally compact, the direct
approach—which is how we proceeded—consists in picking an arbitrary
point $z \in Z$, an arbitrary open neighborhood V of z, and to find a
compact (saturated) neighborhood Q of z included in V. The main
difficulty is in finding enough compact saturated sets Q.

Previous work has shown that $V_{\le 1}X$ is stably compact for every
stably compact space $X$, and gives a hint at what compact saturated
subsets of $V_{\le 1}X$ we may need for the task [2, Theorem 39]. (A stably
compact space is a compact, locally compact sober space that is co-
herent, in the sense that the intersection of any two compact saturated
subsets is compact.) Indeed, in that case, the subsets $[Q \ge r]^{\le 1} \stackrel{\text{def}}{=} \{\nu \in V_{\le 1}X \mid \forall U \in \mathcal{O}X, Q \subseteq U \Rightarrow \nu(U) \ge r\}$ form the required family
of compact saturated sets, where $Q$ ranges over the compact saturated
subsets of $X$ and $r \in \mathbb{R}_+$. We might therefore assume that analogous
sets $[Q \ge r] \cdot \stackrel{\text{def}}{=} \{\nu \in \mathbf{V} \cdot \mathcal{X} | \forall U \in \mathcal{O}\mathcal{X}, Q \subseteq U \Rightarrow \nu(U) \ge r\}$ would be compact saturated in $\mathbf{V} \cdot \mathcal{X}$, and could be used to form enough compact saturated neighborhoods of points $\nu \in \mathbf{V} \cdot \mathcal{X}$, when $X$ is locally compact sober—but without coherence (or compactness).

Without using the detour through superlinear previsions, however, it is not completely clear how one might prove that $[Q \ge r] \cdot$ is compact, to start with. It is true that $[Q \ge r] \cdot$ is compact saturated in $\mathbf{V} \cdot \mathcal{X}$, for every compact saturated subset $Q$ of any topological space $X$, and for every $r \in \mathbb{R}_+$. Indeed, $[Q \ge r] \cdot = s_{\mathrm{DP}}^{\cdot}(F)$, where $F$ is the superlinear prevision $h \in \mathcal{L}X \mapsto r \int_{x \in X} h(x) \, du_Q$, as one may check by using Lemma 3.1.

However, the proof we have given needs many more compact saturated subsets, and we have obtained them as $s_{\mathrm{DP}}^{\cdot}(F)$ for superlinear previsions of the form $h \in \mathcal{L}X \mapsto \int_{x \in X} h(x) \, d\sum_{i=1}^n a_i u_{Q_i}$. In general, $s_{\mathrm{DP}}^{\cdot}(F)$ is compact saturated in $\mathbf{V} \cdot \mathcal{X}$, for any superlinear prevision $F$ on $X$. This naturally leads to the following question.

(1) What are the compact saturated subsets of $\mathbf{V} \cdot \mathcal{X}$, for an arbitrary topological space $X$?

In investigating (1), one may restrict $X$ to some interesting subclasses. The answer to (1) is already known for stably compact spaces: the compact saturated subsets of $\mathbf{V}_{\le 1}X$ are exactly the intersections of finite unions of sets of the form $[Q \ge r]^{\le 1}$ given as above [2]. One should also compare this with Prokhorov's characterization of compact sets of probability measures as those sets that are uniformly tight [20], see also the fundamental paper by Topsøe [25]. In the meantime, one should observe that for every topological space $X$, there is a continuous map $r_{\mathrm{DP}}: \mathcal{Q}_{\mathcal{V}}(\mathbf{V} \cdot X) \to \mathbb{P}_{\mathrm{DP}}^{\cdot}X$ such that $r_{\mathrm{DP}} \circ s_{\mathrm{DP}}^{\cdot} = \mathrm{id}_{\mathbb{P}_{\mathrm{DP}}^{\cdot}X}$ and $s_{\mathrm{DP}}^{\cdot} \circ r_{\mathrm{DP}} \le \mathrm{id}_{\mathcal{Q}_{\mathcal{V}}(\mathbf{V} \cdot X)}$ [12, Proposition 3.22], and that those maps define a homeomorphism between $\mathbb{P}_{\mathrm{DP}}^{\cdot}X$ and the subspace $\mathcal{Q}_{\mathcal{V}}^{cvx}(\mathbf{V} \cdot X)$ of convex elements of $\mathcal{Q}_{\mathcal{V}}\mathbf{V} \cdot X$ [12, Theorem 4.15]. Hence the *convex* compact saturated subsets of $\mathbf{V} \cdot X$ are known: they are exactly the sets $s_{\mathrm{DP}}^{\cdot}(F)$, where $F$ is a superlinear prevision on $X$.

## Acknowledgments

I thank the anonymous referee for carefully rereading this paper.

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UNIVERSITÉ PARIS-SACLAY, CNRS, ENS PARIS-SACLAY, LABORATOIRE MÉTHODES FORMELLES, 91190, GIF-SUR-YVETTE, FRANCE.

E-mail address: jgl@lmf.cnrs.fr