Let be a continuous martingale such that for some . Then there exists a random variable such that as both -almost surely and in . The statement for discrete-time martingales is essentially identical, with the obvious difference that the continuity assumption is no longer necessary. Lévy's zero–one law Doob's martingale convergence theorems imply that conditional expectations also have a convergence property. Let be a probability space and let be a random variable in . Let be any filtration of , and define to be the minimal σ-algebra generated by . Then both -almost surely and in . This result is usually called Lévy's zero–one law or Levy's upwards theorem. The reason for the name is that if is an event in , then the theorem says that almost surely, i.e., the limit of the probabilities is 0 or 1. In plain language, if we are learning gradually all the information that determines the outcome of an event, then we will become gradually certain what the outcome will be. This sounds almost like a tautology, but the result is still non-trivial. For instance, it easily implies Kolmogorov's zero–one law, since it says that for any tail event A, we must have almost surely, hence . Similarly we have the Levy's downwards theorem : Let be a probability space and let be a random variable in . Let be any decreasing sequence of sub-sigma algebras of , and define to be the intersection. Then both -almost surely and in . See also Backwards martingale convergence theorem References (See Appendix C) Probability theorems Martingale theory