Let $\mathcal{X}$ be a topological space. An open subset $\mathcal{R}\subseteq\mathcal{X}$ is regular if it is the interior of its own closure. The intersection of two regular open sets is regular. Unfortunately, the union of two regular open sets is generally not regular. Neither is the complement of a regular open set. Thus, if we define $\mathfrak{R}$ to be the family of all regular open subsets of $\mathcal{X}$, then $\mathfrak{R}$ is not a Boolean algebra under the standard set-theoretic operations. However, $\mathfrak{R}$ is a Boolean algebra under slightly different operations. If $\mathcal{Q}$ and $\mathcal{R}$ are regular open subsets of $\mathcal{X}$, then define $\mathcal{Q}\vee\mathcal{R}:=\mathrm{int}\left[\mathrm{clos}(\mathcal{Q}\cup\mathcal{R})\right]$, and define $\neg\mathcal{Q}:=\mathrm{int}(\mathcal{X}\setminus\mathcal{Q})$. Then $\mathfrak{R}$ is a Boolean algebra under the operations $\vee$, $\cap$, and $\neg$.
We can then define a finitely additive measure on $\mathfrak{R}$ in the obvious way: it is a function $\mu:\mathfrak{R}\longrightarrow\mathbb{R}_+$ such that $\mu[\emptyset]=0$ and $\mu[\mathcal{Q}\vee\mathcal{R}]=\mu[\mathcal{Q}]+\mu[\mathcal{R}]$ whenever $\mathcal{Q}$ and $\mathcal{R}$ are disjoint regular open subsets of $\mathcal{X}$. To avoid confusion with the standard notion of measure (defined in terms of disjoint unions), I will sometimes call this a finitely $\vee$-additive measure in what follows.
So far, this is all standard material: the Boolean algebra structure of regular open sets is well-known, and the idea of defining a finitely additive measure on an arbitrary Boolean algebra has been around for a long time. (See, e.g. volume III of Fremlin's books on measure theory for discussions of both.) But I am interested in three rather specific questions.
(1) What is the relationship (if any) between finitely $\vee$-additive measures on $\mathfrak{R}$ and Borel measures on $\mathcal{X}$?
In some simple cases, a Borel measure on $\mathcal{X}$ "induces" a finitely $\vee$-additive measure on $\mathfrak{R}$. For example, let $\mathcal{X}=[0,1]$ (the unit interval) with the usual topology; then the Lebesgue measure induces a finitely $\vee$-additive measure on the regular open subsets of $[0,1]$ in the obvious way. However, not every Borel measure on $\mathcal{X}$ induces a finitely $\vee$-additive measure on $\mathfrak{R}$ in this way (for example, "atoms" generally create problems). Conversely, not every finitely $\vee$-additive measure on $\mathfrak{R}$ seems to arise from a Borel probability measure.
You might think that the issue here is the disconnect between finite additivity and countable additivity. To avoid this, let $\mathfrak{B}$ be the Boolean algebra generated by all open and closed subsets of $\mathcal{X}$ under the standard set-theoretic operations. We can define finitely additive measures on $\mathfrak{B}$ in the standard way (in terms of disjoint unions). Any Borel measure obviously induces a finitely additive measure on $\mathfrak{B}$ (but not conversely). So we could weaken question (1) to the following:
(2) What is the relationship (if any) between finitely $\vee$-additive measures on $\mathfrak{R}$ and finitely additive measures on $\mathfrak{B}$?
Another question has to do with integration. There is a well-developed theory of integration for any finitely additive or countably additive measure defined on any Boolean algebra of subsets with the standard set-theoretic operations. But this doesn't obviously extend to $\vee$-additive measures.
(3) Is there a well-behaved integration theory for finitely $\vee$-additive measures on $\mathfrak{R}$?
Here, by "well-behaved", I mean that the integration operator is defined for some reasonable domain $\mathcal{F}$ of real-valued functions on $\mathcal{X}$ (e.g. all bounded continuous real-valued functions on $\mathcal{X}$), it is linear on $\mathcal{F}$, it is continuous with respect to some reasonable topology on $\mathcal{F}$, and it is increasing relative to the pointwise ordering of $\mathcal{F}$.
Question (3) is closely related to (1) and (2) because clearly, if we could represent a finitely $\vee$-additive measure in terms of a Borel measure (for example), then we could just invoke the standard integration theory for Borel measures to obtain a positive answer to (3). On the other hand, a positive answer to (3) might lead to a positive answer to (1) and/or (2) via some form of the Riesz Representation Theorem.
I have some ideas about how to answer (1), (2) and (3), but I am worried that I am "reinventing the wheel". These seem to be obvious questions, so I would be surprised if someone hadn't already answered them a long time ago. However, I have looked in the obvious places (e.g. I have searched through Fremlin's encyclopaedic texts on measure theory, done keyword searches on MathSciNet, etc.) and I haven't found anything. But this question lies a bit outside my area of expertise, so perhaps I just looked in the wrong place. So I would be very grateful for any pointers to any literature. Also, I have stated the questions when $\mathcal{X}$ is "any" topological space, but it is likely that we need to impose additional hypotheses on $\mathcal{X}$ (e.g. compact Hausdorff) to get a useful answer.
Additional remarks: To illustrate the problem, I have added the following example. Let $\mathcal{X}:=[-1,1]$ with the usual topology. Let $\lambda$ be the Lebesgue measure. Let $\delta$ be the "point mass" at zero. Finally, let $\nu=\lambda + \delta$; then $\nu$ is a Borel measure on $[-1,1]$ (with total mass 3).
Now, pick two positive numbers $\ell$ and $r$ such that $\ell+r=1$. For any regular open subset $\mathcal{R}\subseteq[-1,1]$, we define $\mu[\mathcal{R}]$ as follows:
if $0\in \mathcal{R}$, then $\mu[\mathcal{R}]:=\nu[\mathcal{R}]$.
if $0\not\in\mathcal{R}$, but $(0,\epsilon)\subseteq\mathcal{R}$ for
some $\epsilon>0$, then define
$\mu[\mathcal{R}]:=\lambda[\mathcal{R}]+r$.
if $0\not\in\mathcal{R}$, but $(-\epsilon,0)\subseteq\mathcal{R}$ for
some $\epsilon>0$, then define
$\mu[\mathcal{R}]:=\lambda[\mathcal{R}]+\ell$.
Finally, if none of these cases hold (equivalently, if $0\in
\neg\mathcal{R}$), then define
$\mu[\mathcal{R}]:=\lambda[\mathcal{R}]$.
Let $\mathfrak{R}$ denote the Boolean algebra of regular open subsets of $[-1,1]$. Then it is easy to check that $\mu$ is a finitely $\vee$-additive measure on $\mathfrak{R}$. Intuitively, $\mu$ is obtained from $\nu$ by "splitting" the point-mass at zero into a "left part" ($\ell$) and a "right part" ($r$), which then get allocated to regular open sets which touch 0 "from the left" and "from the right", respectively.
(This example shows that a finitely $\vee$-additive measure on $\mathfrak{R}$ is not just an ordinary measure on the Borel algebra which vanishes on meagre sets.)