= Coin-doubling martingale
{title2=$M_n=2^n\mathbf1_{\{\xi_1=\cdots=\xi_n=1\}}$}
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>.
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