Related to this question

Setup

$(\Omega, F, P)$; probability space

Let $\{F_n\}_{n=0,1,2,\ldots}$ be a filtration, and $X=\{X_n\}_{n=0,1,2,\ldots}$ be an adapted process.In addition, we define$$X^\ast_n = \max_{k\leq n}|X_k|.$$

Then, $X^\ast=\{X^\ast_n\}_{n=0,1,2,\ldots}$ is adapted.

In addition, we want to show that $X^\ast$ is a martingale.

My try

It is sufficient to show that$$\mathbb{E}[X^\ast_{n+1}\mid F_n] = X^\ast_n.$$

I have not been able to deal with $\max$ well, but I think we can generally indicate this policy:$$\mathbb{E}[X^\ast_{n+1}\mid F_n] = \mathbb{E}[\max_{k\leq n+1}|X_k|\mid F_n] = \ldots = \max_{k\leq n}|X_k| = X^\ast_n$$

I am stuck because I cannot transform it properly.