= Solution
Inductively, $X_0,\ldots,X_n$ are <functions> of $\Delta_0,\ldots,\Delta_n$, so $\mathcal F_n\subseteq\sigma(\Delta_0,\ldots,\Delta_n)$. The average $f_n(X_0,\ldots,X_n)$ is therefore $\mathcal F_n$-measurable and independent of $\Delta_{n+1}$. If $X_0,\ldots,X_n$ are integrable, then
$$
\mathbb E|f_n|\le\frac1{n+1}\sum_{j=0}^n\mathbb E|X_j|<\infty,
\quad
\mathbb E|\Delta_{n+1}f_n|=\mathbb E|\Delta_{n+1}|\,\mathbb E|f_n|<\infty.
$$
Starting with $X_0=\Delta_0$, induction proves integrability of every $X_n$. Adaptation holds by definition of the natural <filtration>. <Independence> and zero mean now legitimately give
$$
\mathbb E[X_{n+1}\mid\mathcal F_n]
=X_n+f_n\mathbb E[\Delta_{n+1}\mid\mathcal F_n]=X_n.
$$
Hence \b[the process is a <martingale>]. Only first moments are used; no second-moment or boundedness assumption on the increments is needed.
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