Measure Theory

This course provides a rigorous introduction to measure theory, an essential mathematical framework for modern probability and advanced statistics. Designed specifically for statistics students, the course begins with fundamental concepts such as sigma-algebras, measurable functions, and measure spaces, laying the groundwork for understanding probability in a more general setting. We then explore Lebesgue integration, highlighting its advantages over Riemann integration and its crucial role in probability theory. Key results like the Monotone Convergence Theorem, Dominated Convergence Theorem, and Fatou’s Lemma will be covered in detail, ensuring students develop a strong intuition for limits and integration in measure spaces. The course also delves into absolute continuity and the Radon-Nikodym Theorem, which provide a rigorous foundation for concepts like conditional probability and density functions. By the end of the course, students will have the mathematical tools necessary to tackle advanced probability, stochastic processes, and statistical inference with confidence.