Minimization of divergences on sets of signed measures

We consider the minimization problem of φ-divergences between a given probability measure P and subsets Ω of the vector space MF of all signed finite measures which integrate a given class F of bounded or unbounded measurable functions. The vector space MF is endowed with the weak topology induced by the class F ∪Bb where Bb is the class of all bounded measurable functions. We treat the problems of existence and characterization of the φ-projections of P on Ω. We consider also the dual equality and the dual attainment problems when Ω is defined by linear constraints.