Is $\mathrm{d} \pi (x,y)$ or $\pi(x,y) \mathrm{d}x \mathrm{d}y$ correct in Optimal Transport formulation?

Question

The Kantorovich formulation of optimal transport sometimes looks like this

$$\mathrm{inf} \int_{E \times E} C(x, y) \ d \pi(x, y)$$

and sometimes looks like this

$$\mathrm{inf} \int_{E \times E} C(x, y) \ \pi(x, y) \hspace{1mm} dx \hspace{1mm} dy $$

both being subject to the usual constraints $$\int_E \pi(x, y) dy = \mu(x) \qquad \int_E \pi(x, y) dx = \nu(y)$$

What is the difference, and which is correct?

Answer 1

You get from the first formula to the second if you assume that the measure $\pi$ has a Lebesgue density on $E\times E$, called by abuse of notation again $\pi$. Then $d\pi(x,y)=\pi(x,y)dxdy$. It is however, not a good idea to impose that π should have a Lebesgue density, because in the most important examples, related to Monge's problem the optimal $\pi$ will actually be singular. In short, the first formula is the better.