Martingales and dyadic intervals

Question

I am looking at the following and having a hard time showing that $X_n$ is a martingale. How to do it?

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Answer 1

Let $\mathcal F_n$ be the sigma field generated by the partition $I_{n,k}:1\leq k \leq 2^{n}$. Note that this sigma field is simply formed by all possible unions of these intervals. Also $\mathcal F_n , n=1,2,...$ is an increasing sequence of sigma fields. It is fairly easy to show that $\{EX|\mathcal F_n\}$ is a martingale for any integrable random variable $X$ and any increasing sequence of sigma fields $\{\mathcal F_n\}$. Now verify using definition of conditional expectation that $X_n=E(f|\mathcal F_n)$