I don't know... what's the motivation?

You're welcome, it is a kind of approximation of $C$...

$W_t$ is Brownian motion, which has $\mathbb E W_t=0$.

Do you consider 1,4,1 to be identical to 4,1,1? Are the numbers in non increasing order?

Nice question... We can ask question about the function $g(x)=E(Y\mid X=x)$, defined by $g(X)=E(Y\mid X)$, as long as they don't distinguish $g$ from another function $h$ where $g(x)=h(x)$ for almost all $x$. But in this case if $g(C)=E(Y\mid X=C)$ then for each particular $x$, $\{C\in x\}$ is a particular measure zero sets of $C$'s and therefore forbidden...

It's classical but it's fairly specific... that structure with the sets $P_2$ and $P_3$ is not the most natural, right?

This question should be closed because it already has an answer at stats.stackexchange.com

Nice construction