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- Publisher
- Springer International Publishing
- Copyright
- © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
- ISBN
- 978-3-031-05987-2
- Pages
- 107 –122
- DOI
- 10.1007/978-3-031-05988-9_5
- Publisher site
- See Chapter on Publisher Site
Abstract
[Jean Ville, we are sometimes told, brought martingales into mathematical probability. In what way is this true? To respond thoughtfully, we must try to see probability theory as Ville and his contemporaries saw it in the 1930s and excavate the martingales already hidden in that theory. Then we may see Ville’s contribution as he himself saw it, as the use of a game to cast light on the denumerable probabilities that Émile Borel had introduced in 1909 and that evolved into the measure-theoretic framework for probability used by mathematicians today. We may then also better understand how Ville’s martingales contributed to two other complementary perspectives on mathematical probability: the understanding of randomness in terms of complexity, and the game-theoretic foundation for probability.]
Published: Oct 18, 2022
Keywords: Algorithmic randomness; Jean Ville; Martingale; Game-theoretic probability; Kolmogorov complexity; Measure-theoretic probability; Game theory; Ville’s inequality; Ville’s theorem