I was reading a paper, where they applied a change of measure without having absolute continuity in the following way: $$\mathbb{E}_{P_XP_Y}[f(X,Y)]\geq \mathbb{E}_{P_XP_Y}[f(X,Y)1_\xi] = \mathbb{E}_{P_{XY}}\left[f(X,Y)1_\xi \left(\frac{dP_{XY}}{dP_XP_Y}\right)^{-1}\right],$$ where $1_\xi$ is the indicator function of the support of $P_{XY}$ and we assume that $P_{XY}\ll P_XP_Y$ so $\frac{dP_{XY}}{dP_XP_Y}$ exists.
Intuitively I would say this does not always work. First of all, it depends on the topology you chose (they didn't specify it). Let's say we pick the intuitive topologies (standard if continuous over $\mathbb{R}^n$, discrete measure if over $\mathbb{Z}$ or $\mathbb{N}$,...). Now, I can see lots of counter-examples for general measures (if $\mu$ is discrete and $\nu=\lambda$, i.e., the Lebesgue measure, clearly the indicator function trick does not work, any single point belongs to the support of $\lambda$ but has measure $0$ with respect to $\lambda$ and positive measure with respect to $\mu$).
Is it possible that the constraints induced by the two measures being a joint and the product of marginals makes this trick work and bypass the requirement of absolute continuity? Is there some convoluted counter-example I don't see?