Lecture
This lecture covers the definitions and properties of martingales in probability theory. It starts by defining a filtration as a sequence of sub-fields of a probability space. The concept of a martingale as a discrete-time process with specific properties is explained through examples like the simple symmetric random walk. The lecture also delves into square-integrable martingales and their key properties, such as the conditional expectations and adaptability. The properties of martingales are thoroughly discussed, emphasizing their significance in the context of probability theory.