Let $(X_{1}\times X_{2},\mathcal{F},\mathbb{P})$ be a probabilty space. And let $\mathbb{P}_{X_{1}}$ be the marginal measure over $X_{1}$, that is $\mathbb{P}_{X_{1}}(A)=\mathbb{P}(A\times X_{2})$, for each measurable set $A$ in $X_{1}$.
Let $f: X_{1}\rightarrow X_{2}$ be a measurable function. Let $A=\{(x,y)\in X_{1}\times X_{2}:y=f(x)\}$. Assume that $A\in\mathcal{F}$.
In a lecture note that I was following, I saw the following formula: quoted:
$$\mathbb{P}(A)=\int_{X_{1}}\mathbb{P}(Y=f(t)|X=t)\;\mathbb{P}_{X_{1}}(t). \quad{(\star)}$$
Since the note didn't mention the definition $\mathbb{P}(Y=f(t)|X=t)$ rigorously, I wonder a rigorous definition of this. I guess that: for each $t\in X_{1}$, we may be able to define a measurable function $t\rightarrow\mathbb{P}(Y=f(t)|X=t)$.
If let's say $X_{1}$ and $X_{2}$ are open interval in $\mathbb{R}$ then I would persuade myself with the following guess:
$$\mathbb{P}(Y=f(t)|X=t)=\lim_{\epsilon\rightarrow 0^{+}}\dfrac{\mathbb{P}(\{(x,y)\in X_{1}\times X_{2}:y=f(x),t-\epsilon <x < t+\epsilon\})}{\mathbb{P}_{X_{1}}(\{x\in X_{1}:t-\epsilon<x<t+\epsilon\})}.$$
My question: what is a rigorous definition of $\mathbb{P}(Y=f(t)|X=t)$?