MATH 481 : Real Analysis II

 

This course deepens the foundations of analysis begun in Real Analysis I. The first part treats Riemann integration (definition via upper/lower sums, Darboux’s Theorem, Riemann sums, and the Fundamental Theorem of Calculus). We then study sequences and series of functions, emphasizing uniform convergence, pointwise limits of continuous/ integrable differentiable functions, and power series and their radii of convergence. The second half introduces Lebesgue measure and integration: Borel \sigma–algebras, outer measure, Lebesgue measurable sets and their properties; simple and measurable functions; the Lebesgue integral; the Monotone Convergence Theorem and the Bounded Convergence Theorem; and the relationship between Lebesgue and Riemann integrals.