First, let's consider the distribution series: $\left\{ \mu _m \right\} _{m\geqslant 1}$ and $\left\{ \nu _m \right\} _{m\geqslant 1}$. They are all defined in the bounded subset $\Omega$ of $\mathbb{R}^n$. As finite positive measures, if they respectively satisfy $\mu _m\ll \nu _m$ (absolutely continuous), then the Radon-Nikodym derivative sequence $\left\{ \frac{\mathrm{d}\mu _m}{\mathrm{d}\nu _m} \right\} _{m\geqslant 1}$ can be defined.
Now, I want to ask: If $D_{\mathrm{KL}}\left( \mu _m, \nu _m \right) \rightarrow 0$ or $D_{\mathrm{TV}}\left( \mu _m, \nu _m \right) \rightarrow 0$ or $\chi ^2\left( \mu _m, \nu _m \right) \rightarrow 0$, do these conditions respectively imply $\left| \frac{\mathrm{d}\mu _m}{\mathrm{d}\nu _m} \right|$ is bounded almost everywhere when $m$ is sufficiently large? More specifically, will they all (or a few, or none) cause $\frac{\mathrm{d}\mu _m}{\mathrm{d}\nu _m}\rightarrow 1$ a.e.?
Hint: You may consider constructing counterexamples similar to the comments on almost everywhere convergence and measure convergence in real variable functions, as it does not hold true. However, unlike standard real analysis techniques, the parts involving probability measures must be treated with caution, especially the constraints of absolute continuity.
Of course, you can also consider that it holds after adding certain conditions, such as the radon nikodym derivative being a martingale. There are many broad inspirations.
Recall: In the following process involving integral operations, the symbols $\mu_m$ and $\nu_m$ we use are only abbreviations and are not true measures. In fact, it should be written as the density corresponding to the measure within the integral. My purpose in writing this is to avoid introducing too many tokens as much as possible.
As is well known, the definition of $f$-divergence is $$ D_f\left( \mu _m, \nu _m \right) =\int_{\Omega}{\nu _m\left( x \right) f\left( \frac{\mu _m\left( x \right)}{\nu _m\left( x \right)} \right) \mathrm{d}x}. $$ The two most famous forms among them are Kullback–Leibler divergence $D_{\mathrm{KL}}\left( \mu _m, \nu _m \right) $: $$ D_{\mathrm{KL}}\left( \mu _m, \nu _m \right) =\int_{\Omega}{\mu _m\left( x \right) \ln \frac{\mu _m\left( x \right)}{\nu _m\left( x \right)}\mathrm{d}x}, $$ and Total-Variation divergence: $$ D_{\mathrm{TV}}\left( \mu _m, \nu _m \right) =\frac{1}{2}\int_{\Omega}{\left| \mu _m\left( x \right) -\nu _m\left( x \right) \right|\mathrm{d}x}=\frac{1}{2}\left\| \mu _m-\nu _m \right\| _{L^1\left( \Omega \right)}. $$ In addition, we often use $\chi^2$-divergence: $$ \chi ^2\left( \mu _m, \nu _m \right) =\int_{\Omega}{\frac{\left( \mu _m\left( x \right) -\nu _m\left( x \right) \right) ^2}{\nu _m\left( x \right)}}\mathrm{d}x. $$ A famous inequality is $$ 2D_{\mathrm{TV}}\left( \mu _m, \nu _m \right) \log e\leqslant D_{\mathrm{KL}}\left( \mu _m, \nu _m \right) \leqslant \ln \left( \chi ^2\left( \mu _m, \nu _m \right) +1 \right) . $$ You can find it from the reference "On Relations Between the Relative Entropy and $\chi^2$-Divergence, Generalizations and Applications" by Tomohiro Nishiyama and Igal Sason, 2020.