Real Analysis II — Measure Theory

Measure theory and Lebesgue integration: the modern foundation of probability, functional analysis, and mathematical physics. Replaces Riemann integration with something more powerful.

Course Overview

Textbook

Folland, Real Analysis (or Royden, Real Analysis)

Chapters

TBD

Sections

TBD

Prerequisites

Real Analysis I

Target

Foundation for Functional Analysis, Probability Theory, PDEs

Status

Planned

Chapters will be scaffolded when this course becomes active.

Why This Course

  • Probability — measure theory IS the rigorous foundation of probability

  • Functional analysis — \(L^p\) spaces are the central objects

  • Quantum mechanics — Hilbert spaces require measure-theoretic integration

  • Machine learning — convergence theorems justify gradient methods

  • Signal processing — Fourier analysis in \(L^2\) requires Lebesgue theory