 Transport potential path construction

Anchor a countable dense subset of a closed c-cyclically monotone support. Take the infimum of accumulated differences $c(x_{j+1},y_j)-c(x_j,y_j)$ along chains ending at a variable point. Cyclical monotonicity bounds this potential below on the support projection. Its cost transform gives the other potential, with equality on the support. Continuity makes both potentials upper semicontinuous and thus Borel.