TITLE: Can a martingale converge in $L^1$ but not almost surely?
QUESTION [8 upvotes]: In general, almost sure convergence and convergence in $L^1$ are independent modes of convergence in the sense that neither implies the other. For martingales, at least one implication, namely that almost sure convergence implies convergence in $L^1$ remains false. How about the other one? Does convergence of a martingale in $L^1$ imply almost sure convergence?

REPLY [4 votes]: Another, somewhat shorter proof: As $L^1$ convergence is equivalent to uniform integrability and convergence in probability, almost sure convergence follows from the convergence theorem for uniformly integrable martingales.