This is a question from Villani's Topics in Optimal Transportation. Let's define $\mu, \upsilon$ probability measures on some measure spaces $X, Y$ respectively, and $\pi$ a probability measure on the product space $X \times Y$.
In the Introduction chapter, pg. 2, there is an intuitive assertion that for a transference plan $\pi \in P(X \times Y)$ to be admissible, it must verify
$$\int_{Y} d\pi(x,y) = d\mu(x) \, \text{and, similarly} \, \int_{X} d\pi(x,y) = d\upsilon(y)$$
which can be more rigorously expressed as:
$$\pi(A \times Y) = \mu(A) \, \text{and} \, \pi(X \times B) = \upsilon(B)$$
for all $A \subset X, B \subset Y$ measurable subsets, is equivalent to
$$\int_{X \times Y} [\phi(x) + \psi(y)] d\pi(x,y) = \int_{X} \phi(x) d\mu(x) + \int_{Y} \psi (y) d\upsilon(y)$$
for all $\phi, \psi$ in a suitable class of test functions.
My question has two parts:
- Where does this result come from? Does it stem from the fact that
$$\pi(A \times Y) = \mu(A) \, \text{and} \, \pi(X \times B) = \upsilon(B)$$ is equivalent to
$$\int_{X \times Y} \mathcal{X}_{A \times Y}(x,y) d\pi(x,y) = \int_{X} \mathcal{X}_{A}(x) d\mu(x) \, \text{and} \, \int_{X \times Y} \mathcal{X}_{X \times B}(x,y) d\pi(x,y) = \int_{Y} \mathcal{X}_{B}(x) d\upsilon(x)$$
- How do we know what class of test functions to choose? Villani says that the natural set of admissible test functions is $L^1(d\mu) \times L^1(d\upsilon)$, which he says is equivalent to $L^{\infty}(d\mu) \times L^{\infty}(d\upsilon)$ (I don't see this either), but he also says that the class can be narrowed down in certain situations.