Nine fully built, single-level courses covering measure, integration, convergence theorems, differentiation, BV/AC, decomposition, and Hilbert/Fourier techniques—optimized for Western higher education.
Complete track: 9 courses
Advanced rigor
Courses
RA-1
Measure Foundations & Outer Measure
Available
Outer measure, Carathéodory criterion, regularity, and measurable sets built from rectangle coverings.
AdvancedPractice jump inside6-10 hours
Outer measure
Carathéodory measurability
Regularity
Null sets
RA-2
Measurable Functions & Approximations
Available
Simple approximations, Egorov/Lusin, stability of measurability, and almost-everywhere control.
AdvancedPractice jump inside7-10 hours
Simple ladders
Egorov
Lusin
a.e. convergence
RA-3
Lebesgue Integration Core & L1
Available
Nonnegative integrals, absolute integrability, L1 geometry, and comparison with Riemann approaches.
AdvancedPractice jump inside8-12 hours
Simple integrals
Fatou
Monotone convergence
Absolute integrability
RA-4
Convergence Theorems: Fatou, MCT, DCT
Available
Limit–integral exchange, uniform integrability heuristics, and sharp counterexamples.
AdvancedPractice jump inside7-11 hours
Fatou lemma
MCT
DCT
Uniform integrability
RA-5
Product Measures & Fubini–Tonelli
Available
Sections, measurability on products, Tonelli for nonnegative functions, and Fubini with integrability.
AdvancedPractice jump inside7-11 hours
Product measures
Tonelli
Fubini
Iterated integrals
RA-6
Differentiation of Integrals & Maximal Control
Available
Hardy–Littlewood maximal function, differentiation a.e., Vitali covering and density theorems.
AdvancedPractice jump inside7-12 hours
HL maximal
Differentiation a.e.
Density points
Vitali covering
RA-7
Bounded Variation & Absolute Continuity
Available
Jordan decomposition, variation measures, absolute continuity, and the Lebesgue Fundamental Theorem of Calculus.
AdvancedPractice jump inside7-11 hours
BV functions
Jordan decomposition
Absolute continuity
FTC (Lebesgue)
RA-8
Lebesgue Decomposition & Radon–Nikodym
Available
Mutual singularity vs absolute continuity, Radon–Nikodym derivatives, and measure decomposition.
AdvancedPractice jump inside7-11 hours
RN derivative
Singular part
Absolutely continuous part
Applications to probability
RA-9
Lp Spaces & Duality
Available
Lp spaces, Hölder and Minkowski inequalities, completeness, and duality theory including Riesz representation.