First we sample from a probability space $\Omega$ and observe the value of $X$, then we sample $Y$ from a probability space $\Omega_X$.

In terms of measure theory, what is $Y$?

Let $X: \Omega \to I$ be a random variable. For each value $x$ of $X$, there is a random variable $Y_x : \Omega_x \to \mathbb{R}$. $Y$ is the value of $Y_x$.

I think $Y$ is a random variable, in other words, a measurable function $Y: S \to \mathbb{R}$ from a probability space $S$. However, what is $S$?

One of the possibilities of $S$ is the set $\Omega \times \bigcup_{x\in I}\Omega_x$ but is this a probability space?

  • $\begingroup$ It sounds like you are over-thinking this. It is not clear from your description if you want $Y$ to be a deterministic function of $X$, meaning $Y=f(X)$ for some function $f$, or if you want $Y$ to have a particular condititional distribution given $X=x$, that is $P[Y\leq y|X=x]$. Either way, there is only one probability space $S$ and $X:S\rightarrow\mathbb{R}$ and $Y:S\rightarrow\mathbb{R}$. The space is whatever is relevant for the system that generates the random vector $(X(s), Y(s))$. Minimally you could make the space $S = \{(x,y) \in \mathbb{R}^2\}$. Or $S = \Omega \times \mathbb{R}$. $\endgroup$ – Michael May 15 at 1:40

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