This note records a common threshold/surplus decomposition for single-threshold stopping rules in the classical prophet inequality. The same decomposition is used to certify several deterministic thresholds, including the median, half-mean, and balanced-surplus thresholds, and to give an averaged certificate for randomized thresholds distributed as the maximum.

翻译：本注释记录了经典先知不等式中单阈值停止规则的通用阈值/盈余分解方法。该分解方法用于验证若干确定性阈值（包括中位数阈值、半均值阈值和平衡盈余阈值），并给出了按最大值分布随机化阈值的平均化验证方法。