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This paper is a continuation of our earlier works [7] and [9] on the fractal structure of expanding and subexpanding meromorphic functions of the form  F  =  H  ° exp °  Q , where  H  and  Q  are non-constant rational maps. Under some assumptions on the forward trajectories of asymptotic values and critical values of  F  we construct a conformal measure  m  for the projection  f  of  F  onto an infinite cylinder  P . Let  J   r  f   consist of all points whose trajectory returns infinitely often to some compact set whose intersection with the postsingular set is empty and  h  = HD( J   r  f  ) be the Hausdorff dimension of  J   r  f  . We prove that  h  ∊ ( s ,2) ( s  will be defined in Corollary 3.4) and an  h -dimensional Hausdorff measure H  h   of  J   r  f   is finite. We construct a unique Borel probability  f -invariant measure μ on  J   r  f  . The measure μ is ergodic and equivalent to  m . We also show that  h -dimensional packing measure of  J   r  f   is locally infinite at every point of this set.

Conformal measures for non-entire functions with critical values eventually mapped onto infinity