Define a probability space $(\Omega,\cal F,\Bbb P)$ and a $\cal F$ measurable random variable $X$, the conditional expectation given a sub $\sigma$-algebra $\cal F_0 \subseteq \cal F$ is a random variable $X_0=\Bbb E(X|\cal F_0)$ satisfying the following two conditions:
- $X_0$ is $\cal F_0$ measurable.
- For any $E\in \cal F_0$, it holds that $\int_E {X} d\Bbb P =\int_E {X_0} d\Bbb P$.
My confusion is that, since $X_0$ is a random variable, it must be a function defined on a sample space, then what is the sample space for $X_0$? Further the sample space must be equipped with a $\sigma$-algebra to form a measurable space, so what is the $\sigma$-algebra? Since $X_0$ is $\cal F_0$ measurable, so I suppose the $\sigma$-algebra associated with the sample space is $\cal F_0$?
Thank you!