I'm a bit confused on some notation used in Arnold's book Random Dynamical Systems and I was wondering if someone may have seen this before. He defines a measure through conditional expectations. Namely, on p.23
$$\mathbb{E}(\mu \ | \ \mathcal{F})_\omega$$
is said to be the conditional expectation of $\mu$ with respect to the $\sigma$-algebra $\mathcal{F}$. How may one express this measure? Should I think of it as some integral? I'm not too familiar with defining measures in this way.
This is essentially a matter of unpacking the definitions and formulas given on pp.22-23. Below is one account of doing this:
If $\mu:\mathcal{F}\otimes\mathcal{B}\to [0,1]$ is a probability measure on $\Omega\times X$ that pushes forward to $\mathbb{P}:\mathcal{F}\to [0,1]$ under the projection map $\pi: \Omega\times X\to \Omega$, then by definition the disintegration $\mu_\bullet$ of $\mu$ is a measurable map $\Omega\to \text{Prob}(\Omega\times X, \mathcal{F}\otimes \mathcal{B})$ such that for any $\omega\in\Omega$, $\mu_\omega$ is a probability measure supported on the fiber $\{\omega\}\times X$.
If $\mathcal{E}\leq\mathcal{F}$ is a sub-$\sigma$-algebra, then $\mathcal{E}\otimes\mathcal{B}\leq \mathcal{F}\otimes\mathcal{B}$. Hence we may consider $\nu=\left.\mu\right|_{\mathcal{E}\otimes\mathcal{B}}:\mathcal{E}\otimes\mathcal{B}\to[0,1]$, which pushes forward to $\mathbb{Q}=\left.\mathbb{P}\right|_{\mathcal{E}}:\mathcal{E}\to[0,1]$. The conditional expectation of $\mu$ w/r/t $\mathcal{E}$ is then by definition the associated disintegration $\nu_\bullet$:
$$\nu_\bullet=\mathbb{E}_\mathbb{P}(\mu_\bullet|\mathcal{E})_\bullet: \Omega \to \text{Prob}(\Omega\times X, \mathcal{E}\otimes \mathcal{B}).$$
(Note that Arnold uses a subscript dot for a measure to denote the disintegration.)
The notation is meant to be evocative of the formula
$$\forall B\in\mathcal{B}, \forall \omega\in_{\mathbb{P}} \Omega: \nu_\omega(B) = \mathbb{E}_\mathbb{P}(\mu_\bullet|\mathcal{E})_\omega(B) = \mathbb{E}_\mathbb{P}(\mu_\bullet(B)| \mathcal{E})(\omega).$$
Here we identify $\mathcal{B}$ with the appropriate fiber $\sigma$-algebra (Arnold uses a slightly different formalism and considers a disintegration to be a map $\Omega\times\mathcal{B}\to[0,1]$; so in the book there is no need for such an identification). Note that given any measurable subset $B\in\mathcal{B}$, the function $\mu_\bullet(B):\Omega\to [0,1]$ is $\mathcal{F}$-measurable, hence in $L^1(\Omega,\mathcal{F},\mathbb{P})$; thus we may refer to the standard theory of conditional expectation to get that $\mathbb{E}_\mathbb{P}(\mu_\bullet(B)| \mathcal{E})\in L^1(\Omega, \mathcal{E},\mathbb{Q})$.