Non-real-valued random variables.

Question

Suppose that $X$ and $Y$ are random variables that take their values in a non-real space, for example in a topological space $\mathcal {U}$.
Is it possible, in that case, to talk about $P[X\in U\mid Y]$, where $U\subset\mathcal {U}$? Any possible definition?

Answer 1

Certainly. The usual definition of conditional expectation defines $P(A|\mathcal G)$ for any event $A$ and any sub$-\sigma-$ algebra $\mathcal G$. Take $A=X^{-1}(U)$ and $\mathcal G=\sigma (Y)=(\{Y^{-1}(V):V \text {Borel}\})$. The Borel $\sigma-$ algebra of a tpological space is the one generated by open sets.