This is probably me being silly, but isn't the second definition just the first one with $E_\Omega=L^1(\Omega)$ for all $\Omega\in \mathcal{A}$? Is there a difference beyond the space in which you take the limit?

What's your definition of a RKHS? I think the usual one is just "a Hilbert space of functions on a set whose evaluation functionals are norm continuous", no?

What happens when $\mathcal{X}=\mathbb{Z}$ and $\mathcal{H}=l^2$?

Just a suggestion, but I think you might want to think about what you're taking as the codomains of $\partial$ and $\bar \partial$, and in what sense you want to be taking derivatives (assuming that's what $\partial$ and $\bar \partial$ stand for that is).

NB: I'm using $\mathrm{limsup}_{y\to x}(*)$ to mean $\mathrm{lim}_{r\to 0}(\sup_{y\in \mathrm{Ball}(x,r)\setminus\{x\}}(*))$

This isn't my area, so I do apologise if I'm speaking out of turn, but... the construction you introduce To make it possible to talk about volumes doesn't seem at all to me like a natural step from what you're talking about at the start of your question. Is your main interest in evolution of volumes in $\mathbb{R}^d$ or are you more interested in abstract analogs of the Liouville trace formula?