Approximation via step functions through the conditional expectation on a sigma-field?

Question

Suppose that $B$ is a Borel sigma field on $[0,1]$ and that we are working with Lebesgue measure. Also, suppose that $B_n$ is the smallest sigma field containing the intervals $I_j = [j2^{-n}, (j+1)2^{-n}]$ for $0 \leq j \leq 2^n -1$. Let $g$ be a function defined as $g : [0,1] \to \mathbb{R}$ where $E(|g|^2) < \infty$. I am trying to write out $E( g | B_n)$ in integral form and to argue that $E(g | B_n)$ will be the mean value on each interval. Does anyone have any idea how I can even start? thanks