So far, I have come across some definition of product measure like this one:
$$d\pi(x,y) = d\mu(x)\delta(y),$$
where $\pi$ is a product measure on the produt space $(X,Y)$, and $\mu$ is a measure on $X$.
But I don't know what $\delta$ means. Is it the delta dirac function? If yes, what the integral
$$\int_{A \times B} f(x,y) d\pi(x,y) = \int_A \int_B f(x,y) \delta(y) d \mu(x)$$
should be, provided that $A$ is a measurable set in $X$, $B$ is measurable set in $Y$ and $f$ is a nonnegative measurable function in $X \times Y$?
Actually, I'm reading about optimal transport, and I came across it in the Villani's book "Topics in Optimal Transportation". Here is the picture:
Any help is highly appreciated.