Functions $f$ such that $A$ and $f(A,B)$ are independent Assume that $A$ and $B$ are two dependent random variables. Are there any results on functions $f$ such that 
$C =f(A, B)$ and $A$ are independent? 
For example, it can easily be shown that  $A$ and $C = F_{B|A}(B, A)$ are independent where $F_{B|A}(., .)$ is the conditional CDF function of $B$, given $A$. (I am using the conditional CDF as a function of two random variables). This is called the Darmois decomposition in the signal processing literature.
So $A$ and $h\Big(F_{B|A}(B, A)\Big)$ are also independent where $h$ is a measurable function. 
Are there any other functions such that $C =f(A, B)$ and $A$ are independent? 
 A: I must say that I don't quite understand your notation. There is a complete description of the independent complements in the sense you are asking, but I prefer to formulate it in somewhat different terms, namely in the language of Lebesgue spaces and their measurable partitions due to Rokhlin. The base probability space in this problem is the joint distribution of $A$ and $B$; let us denote this space by $(X,m)$. Then the random variable $A$ is a function on the space $X$. By one of Rokhlin's theorems, if the distribution of $A$ and almost all conditional distributions of $B$ are purely non-atomic, then the space $(X,m)$ can be identified with the unit square endowed with the Lebesgue measure, and $A$ with the projection of this square onto the first coordinate. Then in this setup you are asking about functions $C$ on the unit square endowed with the Lebesgue measure which are independent of the first coordinate. They are all of the form
$$
C(x_1,x_2) = \phi ( \Psi^{x_1} (x_2)) \;,
$$
where $\Psi^{x_1}$ is a measurable family of automorphisms of the unit interval, and $\phi$ is a measurable function on the unit interval.
