Game-Theoretic Foundations for Probability and Finance

8Zero‐One Laws

If an event depends on a sequence of random variables, and is its probability after are known, then, almost surely, tends to 1 if happens and to 0 if not. More generally, if an integrable random variable is a function of , and is its expected value after are known, then tends almost surely to . This is Lévy's classical zero‐one law, roughly as he stated it in 1937 [245, section ...