Minimization of φ-divergences on sets of signed measures

We consider the minimisation problem of phi-divergences between a given probability measure P and subsets Omega of the vector space M-F of all signed measures which integrate a given class F of bounded or unbounded measurable functions. The vector space M-F is endowed with the weak topology induced by the class F boolean OR B-b where B-b is the class of all bounded measurable functions. We treat the problems of existence and characterization of the phi-projections of P on Omega. We also consider the dual equality and the dual attainment problems when Omega is defined by linear constraints.