Foundations of advanced analysis · Graduate course · Staged public deployment
Size, integration, and limits, made rigorous.
Measure theory is the rigorous foundation of modern analysis, probability, and functional analysis. Beginning from countable set operations and the axioms of sigma-algebras and measures, this course constructs Lebesgue measure through the Caratheodory extension, builds the Lebesgue integral and its convergence theorems, develops product measures, differentiation, and signed measures, proves the Radon-Nikodym theorem and Lp duality, and closes with the probability and Fourier structures that measure theory makes possible.
Where this course sits
What you will master
Assign a rigorous notion of size to sets far beyond intervals, and understand exactly which sets can and cannot be measured.
Build the Lebesgue integral from simple functions and wield the monotone convergence, Fatou, and dominated convergence theorems with confidence.
Prove Radon-Nikodym and Lp duality, then read probability, densities, and Fourier analysis as measure theory in disguise.
The syllabus
Each chapter opens with visual investigations, states every definition and theorem with complete proofs, and closes with thirty exercises, each carrying three progressive hints and a full correction.
Set operations, indexed families, inverse images, countability, and the limsup and liminf of sets. Extended real numbers, indicator functions, and the proof techniques on which measurable structures rest.
Free · Open chapterAlgebras and sigma-algebras of sets, generated sigma-algebras, and the Borel sigma-algebra. Trace, initial, and final measurable structures, and measurable spaces as information structures.
Built · Not yet releasedFinite, sigma-finite, probability, counting, and Dirac measures. Continuity from below and above, subadditivity, null sets, completeness, and the completion of a measure space.
Built · Not yet releasedOuter measures and Caratheodory measurability, Caratheodory's theorem, the construction of measures from premeasures, extension principles, and uniqueness under sigma-finiteness.
Built · Not yet releasedIntervals, length, area, and volume. Lebesgue outer measure and measurable sets, translation invariance, scaling, regularity, Borel versus Lebesgue sets, null sets, and Cantor-type examples.
Built · Not yet releasedEquivalent characterizations of measurability, extended-real-valued functions, operations preserving measurability, pointwise limits, simple functions, positive and negative parts, and almost-everywhere equivalence.
Built · Not yet releasedIntegrals of simple functions, approximation by increasing simple functions, the Lebesgue integral of nonnegative functions, monotonicity, additivity, Fatou's lemma, and the monotone convergence theorem.
Built · Not yet releasedProduct sigma-algebras and measurable rectangles, measurability of sections, the product measure, and the theorems of Tonelli and Fubini, with the counterexamples that show why sigma-finiteness and integrability are required.
Built · Not yet releasedThe Vitali covering lemma and the Hardy-Littlewood maximal function, the Lebesgue differentiation theorem, bounded variation, absolute continuity, and the fundamental theorem of calculus for the Lebesgue integral.
Built · Not yet releasedSigned measures, positive and negative sets, the Hahn and Jordan decompositions, total variation, mutual singularity, and integration with respect to signed and complex measures.
Built · Not yet releasedAbsolute continuity and singularity of measures, the Radon-Nikodym theorem and derivative, the Lebesgue decomposition, change of measure, and applications to probability densities.
Built · Not yet releasedPointwise, almost-everywhere, in-measure, Lp, and uniform convergence; Egorov and Lusin; the Riesz subsequence theorem; uniform integrability and the Vitali convergence theorem; and counterexamples separating each mode.
Built · Not yet releasedHolder duality and the Riesz representation of the dual of Lp via Radon-Nikodym, weak convergence, L2 as a Hilbert space, and conditional expectation as an orthogonal projection with Jensen's inequality.
Built · Not yet releasedProbability spaces and random variables as measurable functions, expectation as a Lebesgue integral, independence via product measures, the Borel-Cantelli lemmas, and L2 Fourier series with Parseval's identity.
Built · Not yet releasedFour premium investigations per chapter, on light crisp backgrounds.
Every theorem stated with all hypotheses and proved in full.
Thirty exercises per chapter with three hints and a full correction each.
Full English and French presentation sharing one mathematical corpus.