Let $C = [0, 1]$. Let $P$ be the uniform probability law on $C$. Let $X : [0, 1] → \mathbb{R}$ be a random variable such that $X(t) = t^2$ for all $t ∈ [0, 1]$. For every integer $k > 1$, let $s = 2^{-k}$ , let $$A_k := \{[0, s), [s, 2s), [2s, 3s), . . . , [1−2s, 1−s), [1−s, 1)\}$$, and let $$M_k := E(X|A_k)$$
Show that the increments $M_2 − M_1$, $M_3 − M_2$, . . . are orthogonal in the following sense.
For any $i, j ≥ 1$ with $i \neq j$,$$E(M_{i+1} − M_i)(M_{j+1} − M_j ) = 0$$
The statement that I want to prove looks like the orthogonality of Martingale, so the remaining thing is to show that $M_K$ is a Martingale. However, I really stuck here since I could not even find another sequence of random variables that $M_k$ is adapted to.
Hope you can help me out!
Thank you in advance.