In the book by Jacod and Shiryaev, there is the following definition, which I rephrase here for convenience.
Definition. Let $(\Omega,\mathcal{F},\{\mathcal{F}_t\},\mathsf{P})$ be a filtered probability space and let $(\Omega^{\prime},\mathcal{F}^{\prime},\{\mathcal{F}^{\prime}_t\})$ be an auxiliary filtered space. Let $\mathsf{Q}(\omega,d\omega^{\prime})$ be a transition probability from $(\Omega,\mathcal{F})$ to $(\Omega^{\prime},\mathcal{F}^{\prime})$. We set $$ \overline{\Omega} = \Omega\times\Omega^{\prime},\quad \overline{\mathcal{F}}=\mathcal{F}\otimes\mathcal{F}^{\prime},\quad \overline{\mathcal{F}}_t=\bigcap_{s>t}\mathcal{F}_s\otimes\mathcal{F}^{\prime}_s,\quad \overline{\mathsf{P}}(d\omega,d\omega^{\prime})=\mathsf{P}(d\omega)\,\mathsf{Q}(\omega,d\omega^{\prime}). $$ and call $\left(\overline{\Omega},\overline{\mathcal{F}},\{\overline{\mathcal{F}}_t\},\overline{\mathsf{P}}\right)$ an extension of $(\Omega,\mathcal{F},\{\mathcal{F}_t\},\mathsf{P})$.
It is not crystal clear to me what is meant with a transition probability. I guess that this is the correct definition:
Definition Any application $\mathsf{Q}:\Omega\times\mathcal{F}^{\prime}\rightarrow [0,1]$ is a transition probability if
- for all $\omega\in\Omega$ it holds that $\mathsf{Q}(\omega,\cdot)$ is a probability measure on $(\Omega^{\prime},\mathcal{F}^{\prime})$,
- for all $A\in\mathcal{F}^{\prime}$ it holds that $\mathsf{Q}(\cdot,A)$ is a measurable function on $(\Omega,\mathcal{F})$.
If this is correct, what is the meaning of $\overline{\mathsf{P}}(d\omega,d\omega^{\prime})=\mathsf{P}(d\omega)\,\mathsf{Q}(\omega,d\omega^{\prime})$ ? My guess is that, for any $A\in\mathcal{F}$ and $A^{\prime}\in\mathcal{F}^{\prime}$ we have
$$ \overline{\mathsf{P}}(A,A^{\prime})=\int_{A}\mathsf{Q}(\omega,A^{\prime})\mathsf{P}(d\omega)=\int_{A}\int_{A^{\prime}}\mathsf{Q}(\omega,d\omega^{\prime})\mathsf{P}(d\omega) $$
Is my guess correct?