Doob's Lemma

Doob's Lemma states that certain types of processes, known as martingales, have predictable future behavior based on their past. Specifically, it suggests that if you're observing a fair game (where outcomes are not influenced by previous results), the best guess for the future outcome, at any point, is simply the current value. In essence, it reinforces the idea that, for fair games, prior results don’t give you an advantage in predicting future results; the expected future value will remain equal to the most recent observed value. This principle has applications in finance and probability theory.