Let $(\Omega ,\mathcal F ,P) := \bigl((0,1],\mathcal B((0,1]),u \bigr)$, where $u$ is the Lebesgue measure restricted to $\mathcal B((0 ,1])$. Let $X\colon\Omega\to\mathbb R$ be defined by $X(\omega) := \omega^2$. $\mathcal F_n:=\sigma(\mathcal E_n)$. $\mathcal E_n=\{(\frac{k-1}{2^n} , \frac k{2^n}]\mid 1 \le k \le 2^n\}$
(a)Determine $H^+(\Omega,\mathcal F)$ and $E(X\mid \mathcal F_1)$
(b)for each $n$, determine a version $Y_n$ of $E[X\mid \mathcal F_n]$.
(c)Show that $E[Y_n\mid\mathcal F_m]= Y_m$ a.s for all $m\le n$.
This is a question from my recent homework. Can anyone give me a hint? By $H^+(\Omega,\mathcal F)$, I mean the set of nonnegative measurable functions .