Coin-doubling martingale ($M_n=2^n{1}_{\{{\xi }_1=\cdots ={\xi }_n=1\}}$)

For independent fair Bernoulli variables, this nonnegative martingale starts at one, doubles on every success, and becomes zero permanently on the first failure. Its expectation is always one, but it tends to zero almost surely because the probability of success forever is zero. Thus it satisfies the Martingale convergence theorem while failing convergence in L1. This illustrates how rare large values obstruct uniform integrability.