Measure theoretic versions of linear programming (Q1092811)

Given topological spaces X, Y and a kernel P from X to Y, a measurable function \(h\geq 0\) on X and a measure \(\nu\) \(\geq 0\) on Y give rise to a pair of dual linear programs: \[ (P)\quad \max imize\quad \int _{X}hd\mu \quad subject\quad to\quad \mu \geq 0\quad and\quad \pi P\leq \nu, \] \[ (D)\quad \min imize\quad \int _{Y}gd\nu \quad subject\quad to\quad g\geq 0\quad and\quad Pg\geq h. \] Two main questions are treated: (1) the duality equation \(\sup (P)=\inf (D)\), (2) the existence of optimal solutions \(\mu\) and g to (P) and (D). Applications concern, among others, general marginal problems as defined by a family of mappings \(\phi _ n:\) \(X\to Y_ n\).