a uniformly integrable martingale

Not every martingale settles down. Brownian motion B_t is a martingale, but B_t does not converge as t -> infinity (it oscillates to plus and minus infinity); the exponential martingale exp(B_t - t/2) converges to 0 almost surely while keeping expectation 1, so its limit is not its 'true' value either. Uniform integrability is precisely the extra condition that distinguishes the martingales that are well-behaved at infinity — those that close up into a single terminal random variable and represent it as a conditional-expectation process — from those that escape to the boundary.

A family of random variables is uniformly integrable (UI) if the tails of their expectations are controlled uniformly: sup over the family of E[|X| ; |X| > K] tends to 0 as K -> infinity. A martingale (M_t) is a UI martingale if the collection {M_t : t >= 0} is uniformly integrable. The fundamental theorem is a tight equivalence: a cadlag martingale (M_t) is UI if and only if it converges almost surely and in L^1 to a limit M_infinity, if and only if it is closed, meaning there is an integrable random variable X with M_t = E[X given F_t] for all t (and then one may take X = M_infinity). For such a martingale the family {M_t} is exactly the conditional expectations of M_infinity, and the optional stopping theorem holds at all stopping times, bounded or not.

This is the continuous-time engine behind martingale representation, Levy's upward theorem, and the cleanest forms of optional stopping. The crucial honesty: L^1-boundedness alone (sup_t E[|M_t|] < infinity) gives only almost-sure convergence (Doob), NOT L^1 convergence and NOT closure — exp(B_t - t/2) is L^1-bounded (expectation 1) yet converges to 0, so E[M_infinity] = 0 != 1 = E[M_0]; uniform integrability is exactly what is needed to upgrade a.s. convergence to L^1 convergence and to preserve the mean. A martingale bounded in L^p for some p > 1 is automatically UI; L^1-boundedness is not enough.