This might be a too naive question.
Recall that the Pettis (or weak) integral is a generalization of the Lebesgue integral for topological vector space-valued functions as follows:
For a measure space $X$ and a "good enough" (e.g. locally convex, Fréchet, Banach, etc.), real topological vector space $E$, we say that $f: X \rightarrow E$ is measurable if for any continuous functional $F \in E^*$ the function $F \circ f: X \rightarrow \mathbb{R}$ is measurable ($X$-measurable).
A measurable function $f: X \rightarrow E$ is said to be Pettis (or weak) integrable on a measurable subset $A \subset X$ if for any $F \in E^*$, the function $F\circ f$ is integrable on $A$ and there exists an element $e \in E$ such that \begin{equation} F(e) = \int_A F \circ f, \quad \forall F \in E^*. \end{equation} In this case we say that the integral of $f$ on $A$ is $e$ and we write $\int_A f = e$.
Now the naive question. Since a natural generalization of a topological vector space is an abelian topological group and some good properties and theorems can be generalized as well (as the notion of locally convexity and the Mackey-Arens Theorem), I've been thinking about if this definition can be generalized to abelian topological group-valued functions $f: X \rightarrow G$ by using the dual of characters (Pontryagin dual) of an abelian group $G^\wedge := CHom(G,S^1)$, instead of the dual vector space $E^*$. However, in order to do this, we need first to define an integral of a circle-valued function $f: X \rightarrow S^1$ , and I think this is not trivial even for the case $X = \mathbb{R}$.
Is there any reasonably good integral for these circle-valued functions, or by the contrary the Pettis integral cannot be generalized in a good way to the case of abelian topological groups?