Phase 6
Real Analysis, Measure & Integration
Phase 6 of the Quant Academy curriculum.
6.1Full Lesson
Metric Spaces, Sequences, Completeness, and Compactness
The topology of convergence: distance, Cauchy sequences, why the reals are complete, and the two faces of compactness.
6.2Full Lesson
Measures and Sigma-Algebras
How to assign a consistent notion of size to sets - and why we cannot measure everything.
6.3Full Lesson
The Lebesgue Integral
Integrate by slicing the range, not the domain - and gain functions the Riemann integral cannot touch.
6.4Full Lesson
Convergence Theorems: Monotone Convergence, Fatou, Dominated Convergence
The three permits for swapping a limit and an integral - the reason the Lebesgue integral was worth building.
6.5Full Lesson
Product Measures, Lᵖ Spaces, and the Bridge to Probability
Fubini for iterated integrals, the geometry of \(L^p\), and the punchline: probability IS measure theory.