We begin as always by defining the population probability space.
$$ \big(\Omega, \mathcal{F}, \mathbb{P} \big) $$
On this space, we can define the potential outcome random variables
$$ \tilde{Y}: \{0,1\} \to \Omega \to \mathcal{R} $$
And our parameter of interest
$$ \theta = \int \tilde{Y}_1 - \tilde{Y}_0 d\mathbb{P} $$
Note, we do not need to (although we can) define the treatment variable on this probability space.
We can define the sample probability space as follows:
$$ \big(\mathbb{\Omega}_n, \mathcal{F}_n, \mathbb{P}_n\big) $$
On this space, we can define the potential outcome random variables (recall that they are not explicitly observed), treatment, outcome, and control variables. Note, we allow for an unrestricted control space
$$ \begin{align*}\tilde{Y} &: N \to \{0,1\} \to \Omega_n \to \mathcal{R} \\ D &: N \to \Omega_n \to \{0, 1\} \\ Y &: N \to \Omega_n \to \mathcal{R} \\ X &: N \to \Omega_n \to \mathcal{X}\end{align*} $$
$$ \begin{align*}\theta &= \int \tilde{Y}_1 d\mathbb{P} \\ &= \int \mathbb{E}[\tilde{Y}_1 \vert X] d\mathbb{P} \\ \int _A \mathbb{E}[\tilde{Y}_1 \vert X] d\mathbb{P} &= \frac{1}{\mathbb{P}(D=1\vert A)}\int _A\mathbb{E}[Y \vert X, D=1]d\mathbb{P}\end{align*} $$
This isn’t true