Let &#x03BC;<!-- μ --> t </msub> be the law of a random process X

As I have mentioned, I would not trust every single part of the paper you are referring to, as there a probably some unchecked typos or even mistakes: since it is on arXiv, it may be not peer reviewed yet. But some things I can help you with clarifying. I think it is more formally correct to write $g_t(x)=\frac{\mathrm{d}{\mathrm{\mu <!--\; \mu \; -->}}_t}{\mathrm{d}{\mathrm{\nu <!--\; \nu \; -->}}_t}(x)$ rather that $\frac{{\mathrm{\mu <!--\; \mu \; -->}}_t(x)}{{\mathrm{\nu <!--\; \nu \; -->}}_t(x)}$ since here ${\mathrm{\nu <!--\; \nu \; -->}}_t(x)$ is not even defined, as a measure it is a function of sets, not of points.
One thing that makes is easier to work with RN derivatives is the following: if $\mathrm{\mu <!--\; \mu \; -->},\mathrm{\nu <!--\; \nu \; -->}\ll <!--\; \ll \; -->\mathrm{\lambda <!--\; \lambda \; -->}$ then
$\begin{array}{}\text{(1)}& \frac{\mathrm{d}\mathrm{\mu <!--\; \mu \; -->}}{\mathrm{d}\mathrm{\nu <!--\; \nu \; -->}}=\frac{\mathrm{d}\mathrm{\mu <!--\; \mu \; -->}}{\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}}/\frac{\mathrm{d}\mathrm{\nu <!--\; \nu \; -->}}{\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}}.\end{array}$
Note that it makes total sense symbolically, but still on the left hand side of (1) you have a RN derivative, not any true ratio, whereas on the right hand side you have indeed a usual ratio of (density) functions. You can always find this $\mathrm{\lambda <!--\; \lambda \; -->}$, e.g. take $\mathrm{\lambda <!--\; \lambda \; -->}=\frac{1}{2}(\mathrm{\mu <!--\; \mu \; -->}+\mathrm{\nu <!--\; \nu \; -->})$. For example, it means that
$\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}t}(\log <!--\; \; -->\frac{\mathrm{d}{\mathrm{\nu <!--\; \nu \; -->}}_t}{\mathrm{d}\mathrm{\nu <!--\; \nu \; -->}})=\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}t}(\log <!--\; \; -->\left(\frac{\mathrm{d}{\mathrm{\nu <!--\; \nu \; -->}}_t}{\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}}/\frac{\mathrm{d}\mathrm{\nu <!--\; \nu \; -->}}{\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}}\right))=\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}t}(\log <!--\; \; -->\frac{\mathrm{d}{\mathrm{\nu <!--\; \nu \; -->}}_t}{\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}}-<!--\; -\; -->\log <!--\; \; -->\frac{\mathrm{d}\mathrm{\nu <!--\; \nu \; -->}}{\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}})=\frac{\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}t}\frac{\mathrm{d}{\mathrm{\nu <!--\; \nu \; -->}}_t}{\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}}}{\frac{\mathrm{d}{\mathrm{\nu <!--\; \nu \; -->}}_t}{\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}}}=\left(\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}t}\frac{\mathrm{d}{\mathrm{\nu <!--\; \nu \; -->}}_t}{\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}}\right)\frac{\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}}{\mathrm{d}{\mathrm{\nu <!--\; \nu \; -->}}_t}.$
Here we used two facts. First of all, $\mathrm{\lambda <!--\; \lambda \; -->}$ should be ${\mathrm{\lambda <!--\; \lambda \; -->}}_t$ in general, but since all ${\mathrm{\nu <!--\; \nu \; -->}}_t$ are said to be dominated by the same measure, we were able to pick a single $t$-independent dominating measure $\mathrm{\lambda <!--\; \lambda \; -->}$. Moreover,
$\frac{1}{\frac{\mathrm{d}{\mathrm{\nu <!--\; \nu \; -->}}_t}{\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}}}=\frac{\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}}{\mathrm{d}{\mathrm{\nu <!--\; \nu \; -->}}_t}.$
Some of these equalities are guaranteed to hold only a.s. but again, everything here is dominated by $\mathrm{\nu <!--\; \nu \; -->}$, so that's not a problem. As a result, for every function $f(x)$ it holds that
$\begin{array}{rl}{\int <!--\; \int \; -->}_{X}f(x)\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}t}(\log <!--\; \; -->\frac{\mathrm{d}{\mathrm{\nu <!--\; \nu \; -->}}_t}{\mathrm{d}\mathrm{\nu <!--\; \nu \; -->}}(x)){\mathrm{\nu <!--\; \nu \; -->}}_t(\mathrm{d}x)& ={\int <!--\; \int \; -->}_{X}f(x)(\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}t}\frac{\mathrm{d}{\mathrm{\nu <!--\; \nu \; -->}}_t}{\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}}(x))\frac{\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}}{\mathrm{d}{\mathrm{\nu <!--\; \nu \; -->}}_t}(x){\mathrm{\nu <!--\; \nu \; -->}}_t(\mathrm{d}x)\\ & ={\int <!--\; \int \; -->}_{X}f(x)(\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}t}\frac{\mathrm{d}{\mathrm{\nu <!--\; \nu \; -->}}_t}{\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}}(x))\mathrm{\lambda <!--\; \lambda \; -->}(\mathrm{d}x)\\ & =\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}t}({\int <!--\; \int \; -->}_{X}f(x)\frac{\mathrm{d}{\mathrm{\nu <!--\; \nu \; -->}}_t}{\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}}(x)\mathrm{\lambda <!--\; \lambda \; -->}(\mathrm{d}x))\\ & =\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}t}({\int <!--\; \int \; -->}_{X}f(x){\mathrm{\nu <!--\; \nu \; -->}}_t(\mathrm{d}x)).\end{array}$
Note also that working with this you have to be accurate in something of the kind $b_t(x)\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}t}{a}_t(x)$ which is
$b_t(x){(\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}s}{a}_s(x))|}_{s=t}$
cause otherwise you may get confused. I hope I didn't.