I have seen the following claim made, but not developed.
Given three vector spaces $X, Y, Z$, one could integrate an $X$-valued function $f$ with respect to a $Y$-valued measure $\mu$ by using a bilinear map $\langle \cdot, \cdot\rangle \colon X \times Y \to Z$ to define an integral $\int f \, d\mu$ with values in $Z$.
Many integrals (e.g. Lebesgue, Bochner) satisfy this definition for specific choices of the vector spaces. But I am looking for discussion of the general approach, not a specific instance. Does anyone know of any references in which it is discussed or developed?