A representation for characteristic Functionals of stable random measures with values in Sazonov spaces

We deal with a Sazonov space (X: real separable) valued symmetric  stable random measure  with independent increments on the measurable space (Rk، B(Rk)). A pair (k، μ)، called here a control pair، for which k: X × Rk! R+، μ a positive measureon (Rk، B(Rk))، is introduced. It is proved that the law of  is governed by a control pair; and every control pair will induce such . Moreover، k is unique for a given μ. Our derivations are based on the Generalized Bochner Theorem and the Radon- NikodymTheorem for vector measures.