Let $(\Omega,{\mathcal F},{\mathbb P})$ be a probability space and ${\mathcal G}$ be a sub $\sigma$-field of ${\mathcal F}$. A system $\{p(\omega),A)\}_{\omega \in \Omega, A \in {\mathcal F}}$ is called a regular conditional probability given ${\mathcal G}$ if it satisfies the following conditions:
(i) for fixed $\omega$, $A \mapsto p(\omega,A)$ is a probability on $(\Omega,{\mathcal F})$;
(ii) for fixed $A \in {\mathcal F}$, $\omega \mapsto p(\omega,A)$ is ${\mathcal G}$-measurable;
(iii) for every $A \in {\mathcal F}$ and $B \in {\mathcal G}$, $$ {\mathbb P}(A \cap B)=\int_{B}p(\omega,A){\mathbb P}({\rm d}\omega). $$
We say that the regular conditional probability is unique if whenever $\{p(\omega),A)\}$ and $\{p'(\omega,A)\}$ possess the above properties, then there exists a set $N \in {\mathcal G}$ of ${\mathbb P}$-measure $0$ such that, if $\omega \notin N$ then $p(\omega,A)=p'(\omega,A)$ for all $A \in {\mathcal F}$.
My question is when the uniqueness of the regular conditional probability holds.
These are described on page 13 of the following book: N. Ikeda and S. Watanabe. Stochastic differential equations and diffusion processes, 2nd edn. North-Holland, Amsterdam, (1981).