This question concerns an extension of a previously solved question regarding measure-theoretic definitions of conditional expectation and conditional independence with respect to a $\sigma$-algebra and an event $A$ (see Conditional expectation and independence on $\sigma$-algebras and events). The mentioned thread considers the situation when $A$ has positive probability. I am interested in the special case when $A$ is a null set.
Background:
Let $(\Omega, \mathcal F, P)$ be a probability space and $X,Y$ be $\mathcal F$-measurable random variables. Let $\mathcal G$ be a sub-$\sigma$ algebra of $\mathcal F$. When $A \in \mathcal F$ and $P(A)>0$ we can make sense of conditional independence with respect to an event and $\sigma$-algebra under $P$ in the usual way. Indeed, by defining the probability measure $P_A(\cdot) := \frac{P(A \cap \cdot)}{P(A)}$ on $(\Omega, \mathcal F)$ we make the identification $$ X \perp_P Y | \mathcal G,A \quad \Leftrightarrow \quad X \perp_{P_A} Y | \mathcal G. $$ I.e. the statement on the left is defined to be the statement on the right. We understand in measure theoretic terms what the independence statement on the right means. Likewise, we understand in measure theoretic terms what e.g. $E_{P_A}[X | \mathcal G]$ means.
My question:
I am really interested in the general situation when $A\in \mathcal F$ is a $P$-null set, but to make the problem more concrete I assume in the following that $A = \{Z=z \}$, where $Z$ is a continuous random variable. It then seems natural to make use of a Markov kernel $K$ (assuming that it exists) from $(\mathbb R, \mathcal B)$ to $(\Omega,\mathcal F)$ and define a regular conditional distribution given $Z=z$ as follows: $P_z(\cdot) := K(z,\cdot)$. I wonder if it is correct to make sense of conditional independence with respect to $\mathcal G$ and $Z=z$ as above: $$ X \perp_P Y | \mathcal G,Z=z \quad \Leftrightarrow \quad X \perp_{P_z} Y | \mathcal G. $$ I.e. that conditional independence and expectation is defined in terms of $P_z$. Since $P_z$ is a probability measure on $(\Omega, \mathcal F)$ we can make sense of the conditional independence statement on the right. Likewise, for $X \in L^1(P_z)$ there is a unique $\mathcal G$-measurable variable $H$ such that $E_{P_z}[I_G H] = E_{P_z}[I_G X]$ for each $G \in \mathcal G$. I wonder if this is a way to make sense of $E_P[X | \mathcal G, Z=z]$.
The obvious limitation here, it seems to me, is that one can not generally assume that such a Markov kernel exists (unless one imposes some special structure on $(\Omega, \mathcal F)$, e.g. that it is a Borel space, which seems undesirable). On the other hand, if the construction I'm thinking of is incorrect then I have no clue as to how objects like $E_P[X | \mathcal G, Z=z]$ are defined in general! Of course, if $\mathcal G = \sigma(L)$ for some random variable $L$, we are really talking about objects like $E_P[X | L, Z=z]$, which statisticians work with and manipulate all the time.