completely-increasing-bijection

For every complete lattice $L$, put $L_*=\{a\in L:a\ne\bot\}$, and let $\mathcal D(L)$ consist of the nonempty subsets $J\subseteq L$ that exclude $\bot$, are upward closed, and are closed under binary meets. Equip $\mathcal D(L)$ with the topology generated by $\{J:a\in J\}$, for $a\in L$. Call a real-valued function $F$ on $\mathcal D(L)$ an abstract observable when it is upper semicontinuous and, for every nonempty family $\mathcal F\subseteq\mathcal D(L)$ and every proper dual ideal $J=\bigcap_{K\in\mathcal F}K$, it satisfies $F(J)=\sup_{K\in\mathcal F}F(K)$. Call $r:L_*\to\mathbb R$ completely increasing when, for every nonempty set $A\subseteq L_*$ and every $a\in L_*$ that is the least upper bound in $L$ of the underlying elements of $A$, $r(a)=\sup_{b\in A}r(b)$. Define

The infima and suprema here are the total real operations: the infimum is $0$ for an empty set or a set unbounded below, and the supremum is $0$ for an empty set or a set unbounded above. The theorem asserts both that every completely increasing $r$ whose range is bounded above and bounded below has $\widetilde r$ an abstract observable with $R(\widetilde r)=r$, and that every abstract observable $F$ has $R(F)$ bounded above and bounded below, has $R(F)$ completely increasing, and satisfies $\widetilde{R(F)}=F$. No orthocomplementation or nontriviality assumption on $L$ is required; when $L$ has only one element, both $L_*$ and $\mathcal D(L)$ are empty, and the assertions concern the corresponding empty-domain functions.