Minimization of divergences on sets of signed measures

We consider the minimization problem of $phi$-divergences between a given probability measure $P$ and subsets $Omega$ of the vector space $mathcal{M}_mathcal{F}$ of all signed finite measures which integrate a given class $mathcal{F}$ of bounded or unbounded measurable functions. The vector space $mathcal{M}_mathcal{F}$ is endowed with the weak topology induced by the class $mathcal{F}cup mathcal{B}_b$ where $mathcal{B}_b$ 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 consider also the dual equality and the dual attainment problems when $Omega$ is defined by linear constraints.