This course introduces the modern foundations of real analysis through the unified framework of measure theory, Lebesgue integration, and Hilbert spaces. Students will first learn how to construct measures from outer measures and develop the concept of measurable sets and measurable functions. The course then develops the Lebesgue integral and the major convergence theorems, emphasizing how the Lebesgue theory extends and refines the classical Riemann approach. Building on this, it examines the connection between differentiation and integration, including the Hardy–Littlewood maximal function, the Lebesgue differentiation theorem, and the notions of absolute continuity and functions of bounded variation. In the second half, attention shifts to Hilbert spaces—particularly the L^2 space—and to orthogonality, projections, and linear operators, culminating in the Riesz representation theorem. Finally, the course extends the discussion to abstract measure spaces, product measures, and signed or complex measures, and introduces modern topics such as the Hausdorff measure and fractal geometry. Throughout the semester, the aim is to cultivate a rigorous and conceptual understanding of analysis, showing how measure theory and integration provide the mathematical foundation for functional analysis, harmonic analysis, and partial differential equations.

Math 311, 312

Real Analysis, Folland
Measure and Integration, Zygmund
Real and Complex Analysis, Rudin

Weeks 1–2: foundations of measure theory, including outer measure, measurable sets, and the construction of the Lebesgue measure.
Weeks 3–5: measurable functions, simple function approximation, and Littlewood’s three principles.
Weeks 6–7: introduce the Lebesgue integral, the Monotone and Dominated Convergence Theorems, and Fubini’s theorem, emphasizing their applications to multiple integration.
Week 8: Midterm Exam
Week 9-10: interplay between differentiation and integration through maximal functions, the Lebesgue differentiation theorem, and the theory of absolutely continuous and bounded variation functions.
Weeks 11–12:Hilbert spaces and L^2 theory, covering orthogonality, projections, and the Riesz representation theorem.
Week 13: examples and applications, including Fourier transforms and PDE connections.
Weeks 14–15: abstract measure spaces, Carathéodory’s extension theorem, product and signed measures, and the Hausdorff measure.
Week 16: Final exam

In-class lecture