Let $\displaystyle (\Omega , F, P)= ([0,1], \beta [0,1], dx)$ , where $\displaystyle \beta$ is a borel set.
$\displaystyle F_n= \sigma ([\frac{k}{2^n}, \frac{k+1}{2^n}, k=0,1,...2^n-2, [\frac{k}{2^n}, \frac{k+1}{2^n}], k=2^n-1)$.
$\displaystyle f(x)=x$.

Then write the explicit formula for $\displaystyle f_n(x)=E(f|F_n)$