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).

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    $\begingroup$ Related: math.stackexchange.com/questions/3526451/… $\endgroup$ – d.k.o. Feb 14 at 16:50
  • $\begingroup$ Thank you for your comment. I have understood as follows: if $\Omega$ is a second-countable space, then the uniqueness holds since ${\mathcal F}$ is countably generated. In particular, if $\Omega$ is a separable metric space, then it holds. Thank you for your advice. (I referred to the following site: math.stackexchange.com/questions/2456416/…) $\endgroup$ – 720773 Feb 14 at 17:35
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    $\begingroup$ Yes, it works when $\mathcal{F}$ is countably generated (e.g. $\mathcal{B}(\mathbb{R})$). $\endgroup$ – d.k.o. Feb 14 at 17:45

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