I have a set of random variables $\mathcal{X}$, where the relations are not numbered (i.e. $\mathcal{X}$ would be of the form $\{X_1, \dots, X_N\}$) but rather have names such as $X_A$ and $X_B$.
I define a random variable $Y = \bigcup\limits_{x_\text{R} \in \mathcal{X}}X_\text{R}$. Here the union works because the realisations of all random variables in $\mathcal{X}$ are sets.
Then, I want to write down the formula for computing $P(Y)$ as follows
$$P(Y) = \sum_{\mathcal{X}} P(Y|\mathcal{X})$$
I want to make it clear that I am summing over all possible realisations of all random variables in $\mathcal{X}$ for computing $P(X)$ but I don't know exactly how to do that.
I came up with this
$$P(Y=y) = \sum_{\substack{ \{x_\text{R} \in \Omega_\text{R} : \text{R} \in \mathcal{R}\}\\ y = \bigcup\limits_{\text{R} \in \mathcal{R}}x_\text{R} }} P(Y=y|\{{x_\text{R}:\text{R}\in\mathcal{R}\}})$$
where $\Omega_\text{R}$ denotes the possible realisations of $X_\text{R}$
but is there a better way?
I am not sure whether I understand things correctly.
Conditional? Then I would rather expect something like $P(Y=y)=\sum P(Y=y\mid\mathcal X)P(\mathcal X)$.
Observe that notation $\bigcup_{X_R\in\mathcal X}X_R$ can be replaced by notation $\cup\mathcal X$.
This because $\cup a$ and $\bigcup_{b\in a}b$ are notations for the same set.
(In set-theory: $x\in\cup a\iff \exists b[x\in b\wedge b\in a]$)
In that context I would think of is something like: $$P(Y=y)=\sum_{\cup A=y}P(\mathcal X=A)$$