We provide a composite version of Ville's theorem that an event has zero measure if and only if there exists a nonnegative 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 nonnegative 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 and that our e-process explodes to infinity on every such violating sequence, while this is provably not achievable with a nonnegative (super)martingale.

翻译：我们给出了Ville定理的一个复合版本：一个事件具有零测度当且仅当存在一个非负鞅，在该事件发生时该鞅趋于无穷大。这是连接测度论概率论与最近由Shafer和Vovk发展的逐序列博弈论概率论的经典结果。我们对Ville结果的推广涉及非负鞅和零测度事件的适当复合推广：这些分别由"e-过程"和一种新的逆资本外测度提供。随后，我们推导了一个仅需随机变量具有有限一阶矩的线交叉不等式，并利用该不等式证明了强大数定律（SLLN）的复合版本。这表明违反强大数定律是一个外测度为零的事件，且我们的e-过程在每一个此类违反序列上都会趋于无穷大，而这一性质对于非负（超）鞅而言是可证明不可实现的。