The Sullivan dictionary provides a conceptual framework to com-pare the actions of Kleinian groups and the dynamics of rational maps.Both settings generate interesting fractal sets (limit sets of Kleiniangroups and Julia sets of rational maps). There is a particularly strongcorrespondence in the context of dimension theory. Restricting to thegeometrically finite setting, in both cases there is a ’critical exponent’which returns the Hausdorff, box, and packing dimensions of the asso-ciated fractal, as well as the Hausdorff, packing, and entropy dimen-sions of the associated conformal measure. We show that, by slightlyexpanding the family of dimensions considered, a much richer theoryemerges. This allows us to draw more nuanced comparisons, and pro-vide novel discrepancies, between the Kleinian and rational map set-tings. This is joint work with Liam Stuart. Letνbe a deterministicmeasure. We wish to find a random measure that solves the equationE(μ) =νwhileμis supported on the Brownian path and is nicelyspread so we can use it as a tool for geometric measure theory of theBrownian path. We describe when the problem can be solved and weprovide a solution. We outline the possible application of the randommeasures. The theory is developed for more general random closed setsthan the Brownian path.