Let $h$ be the Hausdorff dimension of the Julia set of a rational function$T$ with no non-periodic recurrentcritical points and let $m$ be the only$h$-conformal measure for $T$. We prove the existence of a $\sigma$-finite$T$-invariant measure $\mu$ equivalent with $m$. Themeasure $\mu$ is then proved to be ergodic and conservative and we study theset of those points whose all openneighborhoods haveinfinite measure $\mu$. Developing the concept of the inverse jumptransformation we show that the packing and Hausdorffdimensions of the conformal measure are equal to $h$. We also provide somesufficient conditions for Hausdorff and box dimensionsof the Julia set to be equal.