I am reading a paper, but it seems to me that the author uses the terms 'measure' and 'function' interchangeably. For example,
Let us consider a bounded measure $\mu$ so that the measure $(1 + \vert x \vert)\mu$ is also bounded. If moreover $\mu $ belongs to $H^{-1}(\mathbb{R}^2)$, then a unique divergence free vector filed $v$ belonging to $\sigma + L^2(\mathbb{R}^2; \mathbb{R}^2)$ for some stationary smooth vector field $\sigma$ exists so that $\omega(v) = \mu$.
Also, vice versa
Assume that $v_0$ belongs to $E_m $ and that the singular part (with respect to the Lebesgue measure) of $\omega_0$ is a positive measure.
where $E_m = \sigma + L^2(\mathbb{R}^2;\mathbb{R}^2)$ and $\sigma$ is a smooth vector field.
I am wondering if measure and function are used interchangeably by abuse of notation. I haven't read a book where it says about this explicitly. If so, then is the function a Radon-Nikodym derivative with respect to the Lebesgue measure? I'd appreciate any help!