For a probability space $(S,E,P)$ and integrable random variable $X$, if $G = \sigma(X)$ which is generated by a partition $C_1,C_2,\ldots$ with $P(C_i)>0 \ \forall\, i\in{1,2,\ldots}$, then what is $E[X\mid G]$?
Am I correct in that $E[X\mid G]=X$?
My thought was if we let $X(w)=x_i$ for $w \in C_i$. Then
$$E[X\mid C_i]=\frac{1}{P(C_i)}\int_{C_i}X\,dP=\frac{x_i}{P(C_i)}\int_{C_i}dP=x_i\frac{P(C_i)}{P(C_i)}=x_i$$