Transportation flow problems with Radon measure variables

For a multidimensional control problem ${\displaystyle {\left(P\right)}_K}$ involving controls ${\displaystyle u\in L_{\infty }}$, we construct a dual problem ${\displaystyle {\left(D\right)}_K}$ in which the variables ν to be paired with u are taken from the measure space rca (Ω,) instead of ${\displaystyle \left(L_{\infty }\right)\cdot }$. For this purpose, we add to ${\displaystyle {\left(P\right)}_K}$ a Baire class restriction for the representatives of the controls u. As main results, we prove a strong duality theorem and saddle-point conditions.