@DanielFischer Yesterday you said that "For any reasonable integral, you have $$\int_X h(x)\,d\mu \cdot b = \int_X h(x)\cdot b\,d\mu$$ whenever $h$ is a complex valued integrable function on $X$ and $b$ and element of a complex vector space."
Assume you have $a \in \mathcal{A}$, where $\mathcal{A}$ is Banach Algebra and a real valued function $f(x)$ on $[a,b] \subset \mathbb{R}$ then would you have $\int_{a}^{b}f(x)dxa = \int_{a}^{b}f(x)adx$?