A composite generalization of Ville's martingale theorem

We provide a composite version of Ville's theorem that an event has zero measure if and only if there exists a martingale which explodes to infinity when that event occurs. This is a classic result connecting measure-theoretic probability to the sequence-by-sequence game-theoretic probability, recently developed by Shafer and Vovk. Our extension of Ville's result involves appropriate composite generalizations of martingales and measure-zero events: these are respectively provided by ``e-processes'', and a new inverse-capital outer measure. We then develop a novel line-crossing inequality for sums of random variables which are only required to have a finite first moment, which we use to prove a composite version of the strong law of large numbers (SLLN). This allows us to show that violation of the SLLN is an event of outer measure zero in this setting and that our e-process explodes to infinity on every such violating sequence, while this is provably not achievable with a martingale.