As a motivating example, let $Z_i, i=1,...,n$, be iid random elements, T a real-valued function in $(Z_1,...,Z_n)$, and $X_i, i=1,...,n$ are iid that are independent with $Z_i$'s; consider $P(T(Z_1,...,Z_n)\leq t|Z_{(1)}=X_1,...,Z_{(n)}=X_n, X_1=x_1,...,X_n=x_n)$.
It appears intuitive that we can simplify the above expression by manipulating the conditional part: $P(T(Z_1,...,Z_n)\leq t|Z_{(1)}=x_1,...,Z_{(n)}=x_n, X_1=x_1,...,X_n=x_n)$ Then, by independence between $Z$ and $X$, this equals $P(T(Z_1,...,Z_n)\leq t|Z_{(1)}=x_1,...,Z_{(n)}=x_n)$.
However, if the random elements are non-discrete, so that the conditional part has probability zero, then the conditional probability is in general defined using the measure-theoretic version of conditional expectation (via Radon-Nikodym derivative).
Question:
What properties of conditional expectation allow us to perform this kind of operation (swap equivalent events within the conditional part)?
If this operation does not hold in general, then under what condition does it hold? (e.g. does the use of regular conditional probability help?)
Let $f$ be Borel measurable s.t. $E[|f(Z)|]<\infty$. Let $X,Z$ be not necessarily independent. Let $\mathscr{G}=\sigma(\sigma(X=Z)\cup\sigma(X))$ where $\sigma(X=Z)=\{\emptyset,\{X=Z\},\{X\neq Z\},\Omega\}$. Note, by taking the conditional expectation on both sides, $$E[f(Z)|\mathscr{G}]=f(X)\mathbf{1}_{\{X=Z\}}+E[f(Z)|\mathscr{G}]\mathbf{1}_{\{X\neq Z\}}\implies E[f(Z)|\mathscr{G}]\mathbf{1}_{\{X=Z\}}=f(X)\mathbf{1}_{\{X=Z\}}$$ So if $\omega\in \{X=Z\}$ we have $E[f(Z)|\mathscr{G}](\omega)=f(X(\omega))$.