I'm working with a problem with conditional expectation
$\Omega=[0,1)$. $P(A)$ is Lebesgue measure of $A\in\mathcal{B}[0,1)$
$f:[0,1)\to\mathcal{R}$ is bounded. $X(w):=f(w),w\in\Omega$
$\mathcal{G}_N=\sigma\{[\frac{j}{2^n},\frac{j+1}{2^n})|n\leq N,0\leq j \leq 2^n-1\}$
How can I compute $E(X|\mathcal{G}_N),N\geq1$ ? I'm quite confused about the concept of conditional expectation.
I think I need help with an example for $N=1$ and $N=2$ Then I think I can do a induction to solve this.
And I also need some help with $\lim_{N\to \infty} E(X|\mathcal{G}_N)$
Any suggestion would help.