Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X, Y$ random variables, mapping from said space to $(\mathcal{X}, \mathcal{C})$ and $(\mathcal{Y}, \mathcal{D})$, respectively.
The following equation should hold:
$$\mathbb{P}_{X, Y}(E) = \int_{\mathcal{X}}\int_{\mathcal{Y}}\chi_{E}(x, y) \mathbb{P}_{Y\mid X}(dy, x) \mathbb{P}_X (dx),$$
where $E\in \mathcal{C}\otimes\mathcal{D}$.
My question is now how one actually integrates iteratively over the indicator function that takes two arguments? So first I would integrate over $\mathcal{Y}$ but what happens with the indicator function in that integration? And what then happens during the second integration, the integration over $\mathcal{X}$?