Abstract: Distributional overlap is a critical determinant of learnability in domain adaptation. The standard theory quantifies overlap in terms of $\chi^2$ divergence, as this factors directly into variance and generalization bounds agnostic to the functional form of the $Y$-$X$ relationship. However, in many modern settings, we cannot afford this agnosticism; we often wish to transfer across distributions with disjoint support, where these standard divergence measures are infinite. In this note, we argue that tailored'' divergences that are restricted to measuring overlap in a particular function class are more appropriate. We show how $\chi^2$ (and other) divergences can be generalized to this restricted function class setting via a variational representation, and use this to motivate balancing weight-based methods that have been proposed before, but, we believe, should be more widely used.