Integration theory for functions and values with values in topological rings 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?

 A: This is a comment but too long. The leap from the classical case to your general one is, of course, huge but if one takes a more modest one, namely to functions with values in locally convex algebras and their non locally convex analogues, one sees quite clearly what can happen.  To be concrete, we consider the rings of continuous, resp. (equivalent classes of) measurable functions say on the reals, functions from the interval with values in these spaces and finally their integrals with respect to Lebesgue measure).  (This fits into your scheme since we can consider the reals as the subring of  the constant functions).  The first case is well-studied and well-behaved, in particular, continuous or even bounded measurable functions are integrable, but this is no longer true in the non-locally convex case.  The kindergarten reason for this is that while convex combinations of small things are small in the first situation, this can fail in the second one—the standard way of defining an integral (e.g. the Riemann one) is to take convex combinations of function values and then proceed to take a limit.
The two things you lose in going from the l.c.s. case to the t.v.s. are duality and convexity arguments which play a vital role in vector or algebra valued integration. I suppose that what I am trying to say is that without some substitute for these, problems could arise in the very general situation you are envisaging.
