Journal de mathématiques pures et appliquées vol:73 issue:2 pages:191-221
To study finite Borel measures on the Riemann sphere invariant under a rational function R of degree greater than one, we decompose them in an R-invariant component measure supported on the Julia set and a finite number of mutually singular R-invariant component measures vanishing on the Julia set. The latter one can be described easily. For a characterization of the former one, we use a general approach based on a weight function for R on the Riemann sphere. We investigate the relation between weight functions for R and R-invariant Borel probability measures on the Riemann sphere in both directions and discuss how such a measure can be constructed, given a weight function for R.