Alphabeta MathMeasure Theory
Measure theory begins with the domain before the measure: a sigma-algebra is the family of sets on which countable operations stay available, which is the closure property a topology does not supply. This collection establishes that domain. Algebras and sigma-algebras of subsets, measurable spaces, generated sigma-algebras and the minimality that makes the generation well defined come first, with the closure laws, the calculus of generators and a comparison criterion. Pi-systems, lambda-systems and monotone classes make an argument about a generated sigma-algebra tractable, and Dynkin's pi-lambda theorem and the monotone class theorem are proved. The Borel sigma-algebra is then the smallest one containing the open sets, described from intervals, rays, rational boxes, balls and compact sets, shown to commute with the trace on a subspace, and stable under continuous preimages. The transfinite construction by countable ordinals is given with its choice hypothesis stated, and cardinal arithmetic settles the sizes a sigma-algebra can have.
This vocabulary is what the rest of the subject is stated in, and the tracks scaffolded above it reserve it directly. Functional analysis needs it for the Lebesgue spaces and their duality, probability for random elements and the limit theorems, representation theory for Haar and Radon measures, complex analysis for subharmonic functions and boundary values, and differential geometry for integration against a density. The results this library records without proving under measure and integration are discharged when that track is built.
Pathway
The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.
Part 1 · Sigma-algebras and Borel sets
27 pagesCarathéodory extension and Lebesgue measure, measurable functions, integration and convergence, product and signed measures, Radon--Nikodym, the calculus and duality, Riesz--Markov representation, Vitali covering, maximal estimates, differentiation, Hausdorff measure, complex conventions, Riesz--Thorin interpolation and the sharp FTC, with choice and sigma-finiteness hypotheses explicit. Measure preservation then gives Koopman isometries, invariant sigma-algebras, ergodicity and strong mixing criteria; weak mixing means vanishing Cesàro correlations, product ergodicity and no nonconstant eigenfunctions; Poincaré recurrence, first-return maps, Kac's formula, rotations, shifts and Krylov--Bogolyubov lead to the Chacon transformation, ergodic and weakly but not strongly mixing. The dynamics closes with the maximal ergodic inequality, Birkhoff's almost-everywhere theorem and von Neumann's mean theorem, unique ergodicity, Weyl equidistribution, Borel normality and the fair-coin strong law.
- Sigma Algebras and Borel Sets45 results
Sigma-algebras isolate the set operations that remain available under countable constructions.
12 definitions, 12 lemmas, 3 propositions, 17 theorems, 1 corollaryExamples & counterexamples → - Measures and Their Basic Properties50 results
Sigma-algebras provide the measurable domains, while generated sigma-algebras, pi-systems, lambda-systems and Dynkin's theorem provide the machinery used to prove uniqueness.
15 definitions, 4 lemmas, 8 propositions, 16 theorems, 1 corollary, 6 false statementsExamples & counterexamples → Measures on a sigma-algebra, complete measure spaces, continuity from below, the completion of a measure space, and uniqueness of measures agreeing on a sigma-finite…
6 definitions, 7 lemmas, 4 propositions, 10 theorems, 2 corollaries, 5 false statements, 1 remarkExamples & counterexamples →- Lebesgue Measure on Euclidean Space55 results
Assuming countable choice, the outer-measure and Caratheodory-extension machinery from the prerequisite page, together with the Borel sigma-algebra, Heine-Borel compactness…
5 definitions, 15 lemmas, 5 propositions, 20 theorems, 7 corollaries, 3 remarksExamples & counterexamples → An increasing right-continuous function determines a finitely additive interval set function on the half-open algebra, and the Caratheodory extension theorem upgrades that data to a Borel measure on the line.
6 definitions, 2 propositions, 7 theorems, 1 corollary, 5 false statements, 1 remarkExamples & counterexamples →Lebesgue outer measure, translation invariance, Steinhaus's theorem, the Cantor function, perfect sets, and the published choice ledger are the background of this page.
3 definitions, 4 lemmas, 8 theorems, 5 corollaries, 2 examples, 4 counterexamples, 4 false statements, 1 remarkExamples & counterexamples →Measurability is fixed first as a relation between two sigma-algebras, with the extended real line carrying its Borel sigma-algebra and real Euclidean targets carrying the usual Borel structure.
6 definitions, 1 proposition, 11 theorems, 2 corollaries, 6 false statements, 2 remarksExamples & counterexamples →Diameter-power covers lead to Hausdorff measure, its critical dimension, Lipschitz comparisons, Euclidean volume comparison, and exact Cantor and digit-set computations.
5 definitions, 5 lemmas, 4 propositions, 10 theorems, 3 corollaries, 4 remarksExamples & counterexamples →This page builds the Lebesgue integral in the three-stage route fixed by the measure-theory design notes.
9 definitions, 1 lemma, 5 propositions, 18 theorems, 7 corollaries, 7 false statementsExamples & counterexamples →- Modes of Convergence Egorov and Lusin33 results
This page fixes the core dictionary of measure-theoretic convergence modes used later in the library: almost-everywhere convergence, convergence in measure, almost uniform convergence, and convergence in L¹.
7 definitions, 2 lemmas, 2 propositions, 10 theorems, 5 corollaries, 6 false statements, 1 remarkExamples & counterexamples → This page fixes the sigma-finite product-measure route used throughout the rest of the measure-theory track: sections, the measurable-set and functional forms of Tonelli and…
7 definitions, 4 lemmas, 1 proposition, 14 theorems, 2 corollaries, 6 false statements, 5 remarksExamples & counterexamples →This page fixes the signed-measure convention used later in the measure-theory track: a signed measure may take one infinite sign but not both, null sets are defined by…
9 definitions, 2 lemmas, 5 propositions, 12 theoremsExamples & counterexamples →This page is the seam between the elementary Riemann theory and the Lebesgue integral.
1 lemma, 4 theorems, 1 corollaryExamples & counterexamples →This first Lᵖ page fixes the quotient-first convention used everywhere later: elements of Lᵖ(μ) are almost-everywhere classes, not pointwise functions.
6 definitions, 3 propositions, 15 theorems, 3 corollaries, 6 remarksExamples & counterexamples →This page isolates the scalar Radon-Nikodym and Lebesgue-decomposition package in the split the measure-theory design requires: existence of the decomposition, uniqueness of…
4 definitions, 1 proposition, 12 theorems, 2 corollaries, 1 remarkExamples & counterexamples →This page keeps the finite-p density and translation theory separate from the L^∞ endpoint failures, then builds the Euclidean convolution and mollifier package on that exact ledger.
6 definitions, 9 lemmas, 2 propositions, 16 theorems, 2 corollariesExamples & counterexamples →- The Duality of Lᵖ and L^q15 results
This page proves the Lᵖ representation theorem concretely, without moving to abstract dual-space language before the later functional-analysis seam owns it.
1 definition, 5 lemmas, 2 propositions, 3 theorems, 2 corollaries, 2 remarksExamples & counterexamples → Complex Lp spaces are built from measurable components and equality almost everywhere.
2 definitions, 3 lemmas, 4 theoremsExamples & counterexamples →This page distinguishes Radon (open-set inner regular) from all-Borel regularity, constructs the positive C c representation, and then gives the bounded complex C 0 form.
6 definitions, 10 lemmas, 1 proposition, 11 theorems, 1 corollary, 6 false statements, 1 remarkExamples & counterexamples →This page builds the Euclidean maximal-function and differentiation package in the route fixed by the MT-17 design.
7 definitions, 2 lemmas, 3 propositions, 10 theorems, 1 corollary, 1 counterexampleExamples & counterexamples →Finite simple functions give coefficientwise entire families with exact boundary norm powers.
2 lemmas, 1 theorem, 2 corollariesExamples & counterexamples →This page keeps the three classical seams visible. The first block packages the four Dini derivatives and the finite derivative convention. The second proves the rising-sun…
4 definitions, 14 theorems, 6 false statements, 2 remarksExamples & counterexamples →Measure preservation is defined by inverse images and then characterized by invariant integrals.
5 definitions, 2 lemmas, 4 propositions, 5 theoremsExamples & counterexamples →This page isolates the exact regularity class for the Lebesgue fundamental theorem of calculus.
3 definitions, 3 lemmas, 13 theorems, 1 corollary, 1 counterexample, 5 false statements, 4 remarksExamples & counterexamples →Finite measure forces almost every visit to a measurable set to be followed by infinitely many returns.
4 definitions, 7 lemmas, 6 propositions, 10 theorems, 1 corollary, 6 false statementsExamples & counterexamples →Weak mixing has three equivalent descriptions here: vanishing Cesàro averages of absolute centered correlations, ergodicity of the product system, and absence of nonconstant complex eigenfunctions.
3 definitions, 14 lemmas, 3 theorems, 1 false statementExamples & counterexamples →Time averages have two complementary convergence theories. The maximal ergodic inequality yields Birkhoff's almost-everywhere theorem for L¹ observables on sigma-finite…
4 definitions, 5 lemmas, 1 proposition, 9 theorems, 1 corollary, 7 false statementsExamples & counterexamples →