Given a probability space $(S,\mathcal{F},P)$ where $S$ is countably infinite, $\mathcal{F}$ is the full $\sigma$-algebra, and $\forall a \in S, P(a) > 0$, is it reasonable to speak of $P(Y|X)$ when $Y$ and $X$ are constructed with respect to $(y,x)$ with $y,x\in S$ and not uniquely from $x$ or $y$ for $X$ or $Y$ respectively?

For more context, I'm struggling to work through a conversation with my advisor about conditional probability in a probability space defined on something like a computation tree for a nondeterministic Turing machine. I'm interested in discussing the probability of being in state $y$ at time $t+k$ given the machine was in state $x$ at some previous time $t$. The actual probability space and compound events are more involved in their description, but I hope my misunderstanding can be resolved without going into the actual construction. Below is a picture of events $X$ and $Y$ constructed from $x$ and $y$ after inferring $k$. $X$ contains every state visited in $k$ or fewer time steps having visited state $x$ at time $t$, and $Y$ contains every state that could have been visited at time $t$ or later in order to visit state $y$ at exactly time $t+k$.

A representation of two events X and Y rooted by elementary events x and y where Y has positive probability given X.

I would like to speak of $P(Y|X)$, but my advisor has stated that the function $f\colon S \to \mathcal{P}(S)$ that produces some compound event $A$ from $a\in S$ cannot be ambiguous, $\it{i.e.}$ $f(x)$ cannot vary with $y$ (which implies the "hitting time" $k$) or with $k$ directly. If the function is ambiguous, then the claim is that it's fine to use $\frac{P(X\cap Y)}{P(X)}$, but I cannot call it a conditional probability and cannot use any resulting work in probability theory about conditional probabilities without reconfirming the appropriate theorems myself.


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