Multistage random variable

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?

• 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 $S = \Omega \times \mathbb{R}$. – Michael May 15 at 1:40