Is there a definition of an extension of a probability space?

Question

I am reading some notes where the extension of a probability space is used but there is no definition given and I can not find one on the internet.

Is there a clear definition of what an extension of a probability space is?

Answer 1

The probability space $\left(\widetilde{\Omega},\widetilde{\mathcal{F}},\widetilde{\mathbb{P}}\right)$ is called an extension of $\left(\Omega,\mathcal{F},\mathbb{P}\right)$, if it can be constructed as follows: we have an auxiliary measuable space $\left(\Omega^{\prime},\mathcal{F}^{\prime}\right)$ and a transition probability $\mathbb{Q}\left(\omega,d\omega^{\prime}\right)$ from $\left(\Omega,\mathcal{F}\right)$ into $\left(\Omega^{\prime},\mathcal{F}^{\prime}\right)$ such that

$$\widetilde{\Omega}=\Omega\times\Omega^{\prime},\quad \widetilde{\mathcal{F}}=\mathcal{F}\otimes\mathcal{F}^{\prime},\quad \widetilde{\mathbb{P}}\left(d\omega,d\omega^{\prime}\right)=\mathbb{P}\left(d\omega\right)\mathbb{Q}\left(\omega,d\omega^{\prime}\right).$$

Note that if $\mathbb{Q}\left(\omega,d\omega^{\prime}\right)=\mathbb{P}^{\prime}\left(d\omega^{\prime}\right)$ with $\mathbb{P}^{\prime}$ being some probability measure on $\left(\Omega^{\prime},\mathcal{F}^{\prime}\right)$, it becomes the simple case of product space. In the latter case we say the extension is a product extension. For details you can read Jean Jacod's book.