Riemann Integral: Properties and Generalization

This lecture delves into the Riemann integral, discussing characterizations of integrable functions, including conditions for integrability and generalizations of the Riemann integral formula. The instructor explains how to calculate integrals over more complex domains, such as using the Fubini theorem for non-rectangular domains. The lecture also covers the mean value theorem for integrals and the linearity and monotonicity properties of the Riemann integral. Additionally, the instructor demonstrates the application of the integral formula on simple domains and provides insights into extending the formula to higher dimensions. Various properties and corollaries related to integrability and measurability are explored, showcasing the versatility and applicability of the Riemann integral.