Chapter 22: Measure-Theoretic Foundations

Learning Objectives

• Explain why the Riemann integral is insufficient for probability theory and why Lebesgue integration is the right framework
• Construct sigma-algebras and measures, and define measurable functions as the formal notion of random variables
• Define conditional expectation given a sigma-algebra as a measurable random variable and verify its key properties
• State and interpret the Radon-Nikodym theorem and connect it to likelihood ratios in hypothesis testing
• Understand martingales as sequences adapted to a filtration and state the optional stopping theorem