Conditional independence in measure-theoretic terms Let $\Omega$ be a compact Hausdorff space in $\mathbb{C}^n$. Let $\sigma_\Omega$ be the Borel sigma algebra on $\Omega$. Let $\zeta: \Omega\longrightarrow\partial \mathbb{D}$ be a non constant continuous  function. Let $\sigma_{\partial \mathbb{D}}$ be the Borel sigma algebra on $\partial \mathbb{D}$(Unit circle on the complex plane). Now consider the sigma algebra $\sigma_\zeta=\{{\zeta}^{-1}(A): \;A\in \sigma_{\partial \mathbb{D}}\}\subset \sigma_\Omega$.
Now let $f\in L^1(\Omega, \sigma_\Omega, \mu)$ and lets define a new measure $f_\mu$ on $(\Omega,\sigma_\zeta)$ as $f_{\mu}(A)=\int_A f d\mu$. It is easy to see that for $A\in \sigma_\zeta $, ${\mu}(A)=0$ implies $f_{\mu}(A)=0$, i.e $f_{\mu}(A)$ is absolutely continuous with the restriction of $\mu$ to  $\sigma_\zeta$, so by the Radon Nikodym theorem there exists a $g\in L^1 (\Omega, \sigma_\zeta, \mu)$ such that
$\int_A f d\mu =\int_A g d\mu$ for every $A\in \sigma_\zeta$. Lets call this $g$ as the conditional expectation of $f$ and denote it as $E(f|\sigma_\zeta)$.
My question. Can anyone explain me the conditional independence of any two functions $h,k\in L^1(\Omega, \sigma_\Omega, \mu)$ given $\zeta$?
I need to understand  this result in measure theoretic sense. or any reference for the same will be really appreciated.
 A: $\newcommand\si\sigma\newcommand\Si\Sigma\newcommand\Om\Omega\newcommand\F{\mathcal F}\newcommand\C{\mathbb C}$Let $\F:=\si_\Om$. Let $h$ and $k$ be any ($\F,\Si$)-measurable functions from $\Om$ to a set $S$, endowed with a $\si$-algebra $\Si$ over $S$. Let $\si(h):=\{h^{-1}(B)\colon B\in\Si\}$, with similarly defined $\si(k)$.
Then $h$ and $k$ are called conditionally independent given $\zeta$ if
$$P(U\cap V|\si_\zeta)=P(U|\si_\zeta)P(V|\si_\zeta)$$
$\mu$-almost everywhere ($\mu$-a.e.) for any $U\in\si(h)$ and $V\in \si(k)$.
Using standard reasoning, it is easy to show that the following conditions are equivalent to one another:

*

*$h$ and $k$ are conditionally independent given $\zeta$.


*For any $U\in\si(h)$ and $V\in \si(k)$,
$$E(1_U\,1_V|\si_\zeta)=E(1_U|\si_\zeta)E(1_V|\si_\zeta)$$
$\mu$-a.e.


*For any nonnegative Borel-measurable functions $H$ and $K$ on $\C$,
$$E((H\circ h)(K\circ k)|\si_\zeta)
=E(H\circ h|\si_\zeta)E(K\circ k|\si_\zeta)$$
$\mu$-a.e.


*For any bounded Borel-measurable functions $H$ and $K$ on $\C$,
$$E((H\circ h)(K\circ k)|\si_\zeta)
=E(H\circ h|\si_\zeta)E(K\circ k|\si_\zeta)$$ $\mu$-a.e.
