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 $\mathcal{M}_\mathcal{F}$ 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 $ϕ$-projections of $P$ on $Ω$. We consider also the dual equality and the dual attainment problems when $Ω$ is defined by linear constraints.

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