The results of the previous chapter are not the last word about Sullivan conformal measures. Left alone, these measures would be a kind of curiosity. Their true power, meaning, and importance come from their geometric characterizations and their usefulness, one could even say indispensability, in understanding geometric measures on Julia sets, i.e., their Hausdorff and packing $h$-dimensional measures, where, we recall, $h=\HD(J(f))$. This is fully achieved in the present chapter. Having said that, this chapter can be viewed from two perspectives. The first is that we provide therein a geometrical characterization of the $h$-conformal measure $m_h$, which, with the absence of parabolic points, turns out to be a normalized packing measure, and the second is that we give a complete description of geometric, Hausdorff, and packing measures of the Julia sets $J(f)$. Owing to the fact that the Hausdorff dimension of the Julia set of an elliptic function is strictly larger than $1$, this picture is even simpler than for nonrecurrent rational functions.

We first show with proofs the basic and fundamental concepts and theorems from abstract and geometric measure theory. These include, in particular, the three classical covering theorems: 4r, Besicovitch, and Vitali type. We also include a short section on probability theory: conditional expectations and Martingale Theorems. We devote quite a significant amount of space to treating Hausdorff and packing measures. In particular, we formulate and prove Frostman Converse Lemmas, which form an indispensable tool for proving that a Hausdorff or packing measure is finite, positive, or infinite. Some of these are frequently called, in particular in the fractal geometry literature, the mass redistribution principle, but these lemmas involve no mass redistribution. We then deal with Hausdorff, packing, box counting, and dimensions of sets and measures, and provide tools to calculate and estimate them.