= Solution
Put $S_0=0$ and take $\mathcal F_0$ to be the trivial <sigma-algebra>. Each $S_n$ is measurable with respect to the <natural filtration> $\mathcal F_n=\sigma(X_1,\ldots,X_n)$, and
$$
\mathbb E|S_n|\leq\sum_{i=1}^n\mathbb E|X_i|<\infty.
$$
Independence of $X_{n+1}$ from $\mathcal F_n$ makes its <conditional expectation> equal to its <expected value>, which is zero. Hence
$$
\mathbb E[S_{n+1}\mid\mathcal F_n]
=S_n+\mathbb E[X_{n+1}\mid\mathcal F_n]
=S_n+\mathbb E X_{n+1}=S_n.
$$
The measurability, integrability and <conditional expectation> conditions are all verified, so $\boxed{(S_n)\text{ is a martingale for }(\mathcal F_n).}$
Back to article page
