Overview

In this section, we develop the theory of Lebesgue Integration. Unlike the Riemann integral, which partitions the domain of a function, the Lebesgue integral partitions the codomain (range), making it far more powerful for handling complex sets and limit operations.

This framework is essential for defining Expectation in probability theory and for proving the major convergence theorems.

Key Topics

1. The machinery of Lebesgue integration
   • Simple Functions: The building blocks of Lebesgue integration.
   • Bounded Functions: Extending the integral to bounded measurable functions on sets of finite measure.
   • Non-Negative Functions: Generalizing to all non-negative measurable functions.
   • General Functions: Defining the integral for any integrable function.
2. Convergence: The Monotone and Dominated Convergence Theorems.
3. Expectation: Probability-theoretic interpretation of the Lebesgue integral.
4. Moments and Variance: Higher-order statistics defined through integration.
5. Inequalities: Fundamental bounds like Markov’s, Chebyshev’s, and Jensen’s inequalities.