Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 34 / 4 / a

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 ${\mathcal{F}}_n=\sigma (X_1,\dots ,X_n)$, andIndependence of $X_{n+1}$ from ${\mathcal{F}}_n$ makes its conditional expectation equal to its expected value, which is zero. HenceThe measurability, integrability and conditional expectation conditions are all verified, so $\boxed{(S_n)\text{ is a martingale for }({\mathcal{F}}_n).}$