This note investigates core properties of martingales, emphasizing the measure-theoretic formulation of conditional expectation, the martingale transform, and the upcrossing lemma. These results lead to the Martingale Convergence Theorem, which we then apply to study the extinction behavior in Galton--Watson branching processes.

翻译：本文研究了鞅的核心性质，重点探讨了条件期望的测度论表述、鞅变换以及上穿引理。这些结果导向鞅收敛定理，我们随后将其应用于研究Galton-Watson分支过程的灭绝行为。