# When does the uniqueness of regular conditional probabilities hold?

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).

• – d.k.o. Feb 14 at 16:50
• 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/…) – 720773 Feb 14 at 17:35
• Yes, it works when $\mathcal{F}$ is countably generated (e.g. $\mathcal{B}(\mathbb{R})$). – d.k.o. Feb 14 at 17:45