I am curious whether somebody ever tried to generalize the classical theory of Lebesgue integral to functions and measures with values in Hausdorff topological rings.

The generalization of a measure is straightforward: given a topological ring $R$ and a $\sigma$-algebra $\mathcal A$ on a set $\Omega$, define an $R$-valued measure as a function $\mu:\mathcal A\to R$ such that

$\bullet$ $\mu(A\cup B)=\mu(A)+\mu(B)$ for any disjoint sets $A,B\in\mathcal A$;

$\bullet$ $\mu(\bigcup_{n\in\omega}A_n)=\sum_{n\in\omega}\mu(A_n)$ for any sequence $(A_n)_{n\in\omega}$ consisting of pairwise disjoint sets in the algebra $\mathcal A$.

Given a simple $\mathcal A$-measurable function $f:\Omega\to R$ and an $R$-valued measure $\mu$, define the integral $\int f d\mu$ as the (finite) sum $\sum_{y\in R}y\cdot\mu(f^{-1}(y))$.

So, the question:

Is anything known about topological rings $R$ for which the $R$-valued integral can be defined for some reasonably wide class of functions and so-generalized integral has all basic properties of the usual Lebesgue integral?