Alphabeta MathMeasure Theory
The Measure Theory pages develop measurable spaces, sigma-algebras, measures, and integration. They prove generation and extension results, construct Lebesgue measure, and treat measurable functions, simple approximation, convergence theorems, product measures, and Fubini–Tonelli. Further topics include signed and complex measures, Radon–Nikodym and Lebesgue decomposition, Lebesgue–Stieltjes measures, Radon representation, Hausdorff measure, differentiation and maximal functions, and the relation between Lebesgue and Riemann integration. The collection also develops Lp spaces and duality, modes of convergence, measure-preserving systems, recurrence, mixing, and the ergodic theorems of von Neumann and Birkhoff. It also treats nonmeasurable sets, choice assumptions, and the cardinalities of sigma-algebras.
Real Analysis supplies the main completeness and integration foundations, alongside results from Topology, Set Theory, Linear Algebra, Abstract Algebra, and Complex Analysis. Partial Differential Equations, Fourier Analysis, Probability, Functional Analysis, Complex Analysis, Mathematical Statistics, and Representation Theory build on the collection's integration, convergence, and Lp results in their own arguments, including work on operators, random laws, and ergodic theory.
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
28 pagesA total pointwise product for extended-real functions, with no measurability claim, opens the part: the zero rule covers every ordered zero--infinity pair, signs govern infinite factors. Carathéodory extension, Lebesgue measure, measurable functions, integration, convergence, product and signed measures, Radon--Nikodym, calculus and duality, Riesz--Markov, Vitali covering, maximal estimates, differentiation, Hausdorff measure, complex conventions, Riesz--Thorin interpolation, sharp FTC, with choice and sigma-finiteness explicit. Measure preservation gives Koopman isometries, invariant sigma-algebras, ergodicity, strong mixing; weak mixing means vanishing Cesàro correlations, product ergodicity, no nonconstant eigenfunctions; Poincaré recurrence, first-return maps, Kac's formula, rotations, shifts, Krylov--Bogolyubov yield Chacon, weakly but not strongly mixing, closing with the maximal ergodic inequality, Birkhoff and von Neumann ergodic theorems, Weyl equidistribution, Borel normality, the fair-coin strong law.
This page defines a total pointwise product of functions from an arbitrary set into the extended real line The extended real line ℝ̄ = ℝ ∪ {-∞, +∞}, its order, and the arithmetic that is left undefined.
1 definitionExamples & counterexamples- 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 & counterexamplesLebesgue 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 & counterexamplesMeasurability 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 & counterexamplesDiameter-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, 3 remarksExamples & counterexamplesThis 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 & counterexamplesThis 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 & counterexamplesThis page is the seam between the elementary Riemann theory and the Lebesgue integral.
1 lemma, 4 theorems, 1 corollaryExamples & counterexamplesThis 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 & counterexamplesThis 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 corollariesExamples & counterexamplesThis 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^q14 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, 1 remarkExamples & counterexamples Complex Lp spaces are built from measurable components and equality almost everywhere.
2 definitions, 3 lemmas, 4 theoremsExamples & counterexamplesThis 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 & counterexamplesThis 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 & counterexamplesFinite simple functions give coefficientwise entire families with exact boundary norm powers.
2 lemmas, 1 theorem, 2 corollariesExamples & counterexamplesThis 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 & counterexamplesMeasure preservation is defined by inverse images and then characterized by invariant integrals.
5 definitions, 2 lemmas, 4 propositions, 5 theoremsExamples & counterexamplesThis 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, 3 remarksExamples & counterexamplesFinite 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 & counterexamplesWeak 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 & counterexamplesTime 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