Interpretation of $\mathbb P(A|X=x)$ in two ways Let $X:(\Omega,\mathscr A) \to (\mathbb R,\mathscr{B})$ be a random variable between two measurable spaces (the latter being the Borel measurable space over $\mathbb R$). Let $x\in \mathbb R$. Let $\mathbb P$ be a probability on $(\Omega,\mathscr A)$. Assuming $\mathbb P(X=x)\ne 0$ ,I have two interpretations for $\mathbb P(A|X=x)$:
(1) Naive definition: Simply use the elementary definition of the conditional probability:
$$\mathbb P(A|X=x)=\frac{\mathbb P(A\cap[X=x])}{\mathbb P([X=x])}.$$
(2) Official definition of such conditional probability:
$$\mathbb P(A|X=x)=\mathbb E(\mathbb 1_A|X=x)=\varphi(x),$$
where the conditional probability $\mathbb E(\mathbb 1_A|X=x)=\varphi(x)$ is defined via the "factorization lemma", which states that there exists a measurable $\varphi:(\mathbb R,\mathscr{B})\to (\mathbb R,\mathscr{B})$ such that $\varphi(X)=\mathbb E(\mathbb 1_A|X):=\mathbb E(\mathbb 1_A|\sigma(X))$. ($\mathbb E(\mathbb 1_A|\sigma(X))$ is the conditional probability in the usual sense).
The definition in (2) can be found for example in Probability Theory: A Comprehensive Course, by Achim Klenke, page 180-181.
Do the conditional probabilities in (1) and (2) agree (at least almost surely)? If they only agree under some further assumptions, please let me know.

The Naïve definition is undefined when $\Bbb P([X=x])=0$.  Otherwise they agree.
