Let $\mu$ be a Borel measure on the real line $\mathbb{R}$ taking values in a separable Banach algebra $A$. Assume that $\mu$ is such that the total variation measure $|\mu|$ is finite. Let $f$ be a bounded measurable function on the real line $\mathbb{R}$ taking values in a Banach algebra $A$. Is there a general theory that makes sense of integrals of the form


$$
\int\_{\mathbb{R}} f(x)\mu(dx) \in A.
$$


I would be also interested in special cases: $A$ a sub-algebra of $B(H)$, where $H$ -- a Hilbert space or $A$ a commutative algebra.



Integration in Banach algebra
I believe OP's setting is an instance of the more general so called *bilinear integral*, defined for the following data:


* a measurable space $(S,\mathfrak{S})$;
* a function $f: S \to X$, where $X$ is a normed linear space;
* an *additive* measure $\mu: \mathfrak{S} \to Y$, where $Y$ is another normed linear space;
* a bilinear map $\beta: X \times Y \to Z$, where $Z$ is a Banach space;


Then, for $E \in \mathfrak{S}$, one can consider
$$
\int\_E \beta(f(s),\mu(ds))
$$


In OP's setting, the bilinear mapping is given by the multiplication in the Banach algebra.


The canonical reference is *Bartle - A General Bilinear Vector Integral (1955)*, but some further references, that de facto establish it as a theory, are:


* *Dunford - A Bilinear Vector Integral (1975)*
* *Freniche, Garcia-Vazquez - The Bartle Bilinear Integration and Carleman Operators (1998)*
* *Jefferies - Bilinear Integrals and Radon-Nikodym Derivatives*
* *Jefferies - Some Recent Applications of Bilinear Integration*


A generalization of Bartle's integral is obtained and substantially expanded upon in Dobrakov's series of 13 papers titled *"Integration in Banach Spaces"*. There he considers $f: S \to X$, where $X$ is a Banach space, and $\mu: \mathfrak{S} \to BL(X,Y)$, where $Y$ is another Banach space, but this reduces to OP's setting via the adjoint action of the Banach algebra.


**EDIT:** Apologies, forgot to mention that Panchapagesan has a kind of a nice survey/summary on Dobrakov's integral, titled *"On the Distinguishing Features of the Dobrakov Integral (1991)"*, which should help navigate Dobrakov's series of papers. Moreover, Panchapagesan apparently has a whole **book** *"The Bartle-Dunford-Schwartz Integral"* that is relatively recent (from 2000 or so), but I am not sure to what extent it addresses OP's setting, but it might be worth looking into.


