In this machine learning paper, the following lemma is stated (and proven in the Appendix A, cf. page 11):
Lemma A.1 For random variables $X$, $Y$ and function $f(x, y)$ under suitable regularity conditions: $$\mathbb E_{x\sim X, y\sim Y \mid x}[f(x, y)] = \mathbb E_{x\sim X, y\sim Y \mid x, x'\sim X\mid y}[f(x', y)].$$
In their proof, I don't understand the following transition: \begin{align} \mathbb E_{x\sim X, y\sim Y\mid x}\left[f(x, y)\right] = \dots &= \int_{x}\int_{y}P(x, y)f(x, y)\int_{x'}P(x'\vert y)dx'dydx \\[6pt] &= \int_{x}P(x)\int_{y}P(y\vert x)\int_{x'}P(x'\vert y)f(x', y)dx'dydx \end{align} In particular, I'd like to understand why $f(x, y)$, which is integrated over $x$ in the first line, can be integrated over $x'$ in the second line? Thanks a lot!
My interpretation: they meant to write \begin{align} \mathbb E_{x\sim X, y\sim Y\mid x}\left[f(x, y)\right] = \dots &= \int_{x}\int_{y} \int_{x'} P(x, y)f(x, y) P(x'\vert y)dx'dydx \\[6pt] &= \int_{y}\int_{x} \int_{x'}P(y) P(x\vert y)P(x'\vert y)f(x, y)dx'dydx \\[6pt] &= \int_{y}\int_{x}\int_{x'}P(y) P(x\vert y)P(x'\vert y)f(x', y)dx'dydx \end{align} The order of integration doesn't matter by Fubini. The final equality holds because of symmetry: Simply swap the variables $x$ and $x'$.