Durett's proof of Skorokhod’s representation theorem

Question

I have a question on Skorokhod’s representation theorem in Durrett's Probability: Theory and Examples ( https://services.math.duke.edu/~rtd/PTE/PTEv5a.pdf starting at page 347). It says below:

part 1 of proof

part 2 of proof

I don't understand the circled equation (part 2 of proof). It looks like the expectation is taken over $(-\infty,0) \times (0,\infty)$, with probability measure defined in (8.1.1). But the function whose expectation is taken depends on measures induced by points in $\mathbb{R}^2$. What kind of technique is this? Can anyone give me a rigorous (non-intuitive) justification? Any reference is appreciated!