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.

Given that the set $G := \{\omega: X(\omega) = x\}$ has positive probability, by conditional probability definition, $\varphi(X)$ satisfies
$$P(A \cap G) = \int_G \varphi(X(\omega))dP = \int_{\{x\}}\varphi(y)\mu(dy) = \varphi(x)\mu{(\{x\})} = \varphi(x)P[X = x],$$
where $\mu$ is the induced probability measure on $(\mathbb{R}, \mathscr{B})$.
In above, the second equality follows the change-of-variable formula. The third equality follows the definition of integration.
Hence your conjecture is correct, as it should be. 
