I have a paper that I am working on. I wanted to know can a function $\nu$ which is absolutely continuous with respect to measure $\mu$ be represented as an integral where the function is having a factor multiplied in the variable.
From the theory of vector measures, it is clear that, if $(I,\mathscr{F}, \lambda)$ is a finite measure space then Riesz Representation Theorem and Radon-Nikodym Theorem guarantee that there exists a non-negative real-valued measurable function $f$ on $I$ such that for $E \in \mathscr{F}$ we can find \begin{equation*} \nu (E) = \int_{E}f d\lambda \end{equation*}
For example, the function $f$ is having two variables $\{\alpha_1, \alpha_2\}$ and for each variable, there is a factor multiplied which is again a vector $\{r_1,r_2\}$. So, the value would be \begin{equation*} \nu (E) = \int_{E}f (r_1\alpha_1,r_2\alpha_2) d\lambda \end{equation*}
Can someone explain?