Dashboard Preview

This section builds the real-analytic and set-theoretic foundation that Sections 2–11 will consume. Begin with Foundations of the Real Numbers: the field and ordered-field axioms, the least-upper-bound property, and the uniqueness of the complete ordered field establish ℝ as the ambient object on which Lebesgue measure will be built. From your stated background you have limits and continuity for single-valued functions; the field-theoretic material (def-field, def-ordered-field, def-abs-value, def-canonical-natural) is not implied by that background, so read it rather than skim it, since σ-algebras and measures are defined over subsets of structures carrying these operations. Suprema and Infima then converts the least-upper-bound property into the infimum machinery (thm-infimum-property) used whenever you take monotone limits of measures.

Topology of ℝ supplies the vocabulary for Borel sets and for regularity: open and closed sets, closure and interior, limit points, compactness, and the countable decomposition of open sets. These are the exact inputs for σ-algebras and Borel sets in Section 2, so do not skip the closure characterisations or Heine–Borel. Countability and Uncountability is the bridge to the measure-theoretic setting: equinumerosity, Schröder–Bernstein, countable products and unions, Cantor's theorem, and the uncountability of ℝ explain why countable additivity is the right axiom and why non-measurable sets in Section 2 require a choice principle. Note the library gap: countable unions of countable sets assume the Axiom of Countable Choice, and no page here constructs Lebesgue measure on ℝ directly; that begins in Section 2. Landmark results in this section include def-complete-ordered-field, thm-uniqueness-complete-ordered-field, thm-infimum-property, thm-heine-borel-characterisation-r, thm-closure-characterisations-r, def-countable, thm-schroder-bernstein, and thm-r-uncountable.

This section builds the measure-theoretic scaffolding you need for probability, starting only from your real-analysis background (sequence and function limits, continuity, differentiability). Begin with Sigma Algebras and Borel Sets, working through Algebras of subsets, Sigma-algebras, Measurable spaces and measurable sets, and The sigma-algebra generated by a family of sets together with its existence and minimality theorem. The generator machinery then pays off in Dynkin's pi-lambda theorem and The monotone class generated by an algebra equals the sigma-algebra it generates, and in The Borel sigma-algebra of a topological space via Seven generating families for the Borel sigma-algebra on the real line.

Next, Measures and Their Basic Properties introduces Measures on sigma-algebras, Measure spaces, and continuity from below/above, with subadditivity and the first Borel-Cantelli lemma already in a form probability will reuse. Then Outer Measure and the Caratheodory Extension Theorem supplies Outer measures, Carathéodory's theorem, Premeasures on algebras, and the extension and uniqueness theorems, giving the construction route to Lebesgue measure. Finally Non Measurable Sets and the Cost of Choice shows why the axioms are needed: Vitali sets, Bernstein sets, and the failure of translation-invariant measures on all subsets. Landmark definitions and theorems: Sigma-algebras; Measurable spaces; The sigma-algebra generated by a family of sets and its existence/minimality; The Borel sigma-algebra; Dynkin's pi-lambda theorem; The monotone class theorem; Measures on sigma-algebras; Measures on sigma-finite pi-systems uniqueness; Outer measures; Carathéodory's theorem; Premeasures and the Carathéodory extension theorem; Vitali set non-measurability and the no-translation-invariant-measure theorem.

This section builds the Lebesgue measure machinery on $\mathbb{R}^n$ that the probability sections later rely on, starting from limits and continuity in real analysis and ending at measurable functions and the standard modes of convergence. It opens with half-open boxes and their volume, then passes to elementary sets and the well-definedness of elementary volume as a finitely additive, sigma-finite premeasure, which is the input to the Carathéodory construction. Lebesgue outer measure is then defined as the induced outer set function (with the countable-choice hypothesis recorded explicitly wherever it is used), and Lebesgue measurable sets and $\lambda_n$ are introduced via the Carathéodory criterion. From there the section develops the structural results: $\mathcal{L}(\mathbb{R}^n)$ is a complete measure extending elementary volume, every Borel set is measurable, boxes of every face convention have the expected measure, and outer and inner regularity, the Littlewood equivalences, and the agreement of nullity with the covering definitions are established. Metric and geometric properties follow (translation invariance, dilation and linear scaling, uniqueness of the normalised translation-invariant Borel measure, nullity of hyperplanes, and Steinhaus's difference-set theorem). The second page turns to measurable functions: threshold characterisations, the generating-family criterion, closure under arithmetic, lattice and sequential operations, and the explicit increasing simple approximation of nonnegative measurable functions, together with almost-everywhere equality and Doob–Dynkin factorisation. The third page treats modes of convergence — almost everywhere, in measure, almost uniform, and in $L^1$ — and their interrelations, culminating in Riesz's subsequence theorem, Egorov's theorem, Lusin's theorem, and the Vitali convergence theorem. Library gaps: there is no dedicated standalone page for monotone-class/Dynkin arguments or for uniform integrability as its own topic; the relevant material sits inside the listed items.

Landmark items: half-open boxes and volume; elementary sets as finite unions of boxes; well-definedness of elementary volume; elementary volume as a sigma-finite premeasure; Lebesgue outer measure; outer-measure and agreement theorem (under countable choice); Lebesgue measurable sets and $\lambda_n$; completeness of Lebesgue measure; dyadic cubes and the countable disjoint dyadic decomposition of open sets; Borel sets are Lebesgue measurable; measure of a box of every kind; outer regularity; Littlewood's four characterisations; inner regularity by compact sets; agreement of nullity with closed-interval and closed-cube covers; Jordan measurability implies Lebesgue measurability with equal content; translation invariance; dilation and reflection scaling; uniqueness of the normalised translation-invariant Borel measure; linear scaling by invertible maps; nullity of affine hyperplanes and proper subspaces; linear change of variables; Steinhaus difference-set theorem; the generating-family criterion; threshold characterisations of measurability; sequential suprema, infima and limits are measurable; increasing simple approximation of nonnegative measurable functions; Doob–Dynkin lemma; Riesz's subsequence theorem; Egorov's theorem; Lusin's theorem; the Vitali convergence theorem.

This section builds the integration machinery needed before a rigorous measure-theoretic introduction to probability. It assumes only single-variable real analysis: limits of sequences and functions, continuity, and differentiability. Starting from extended-real-valued measurable functions, it develops nonnegative simple functions, the simple integral, and the nonnegative Lebesgue integral over a measurable subset. The convergence theorems are the core: monotone convergence, Fatou's lemma, dominated convergence, and the identification of the indefinite integral as a measure with density. These yield additivity, linearity on L¹, the integral triangle inequality, almost-everywhere criteria, Chebyshev–Markov, Jensen, and absolute continuity of the integral. Two comparison pages relate the Lebesgue and Riemann integrals, including Riemann–Stieltjes versus Lebesgue–Stieltjes, and fix the pointwise extended-real product convention. Two library gaps matter: no standalone Lebesgue-measure-on-R page and no monotone class/Dynkin theorem are supplied, so both are left to other sections; elementary probability is also absent by design, as the target is the measure-theoretic route.

Landmark results include monotone convergence for the integral, Fatou's lemma, a nonnegative integral vanishing exactly on almost-everywhere zero functions, dominated convergence, absolute continuity of the integral, Jensen's inequality for probability measures, the agreement of bounded Riemann integrable functions with their Lebesgue integrals, and the Riemann–Stieltjes/Lebesgue–Stieltjes agreement for continuous integrands.

This section builds the Lp machinery needed for a rigorous measure-theoretic introduction to probability. It begins by fixing the standing objects: conjugate exponents (including the endpoint convention 1/∞=0), the essential supremum used to define the p=∞ size of a measurable function, the class L^p(μ) for 0<p<∞ and the space L^∞(μ) of essentially bounded functions, the null subspace N(μ) of functions vanishing almost everywhere, and the quotient L^p(μ) by almost-everywhere equivalence.

With these in place, the core analytic inequalities and completeness results are supplied, so that the Lp scale is available as a family of complete normed spaces for probability-theoretic work.

This section fuses the product-measure machinery with the Hahn–Jordan and Radon–Nikodym theory, providing the two measure-theoretic pillars on which rigorous probability rests: iterated integration for joint laws and density-based conditioning. On the product side, the section defines measurable rectangles, the product sigma-algebra and its finite iterates, and the horizontal and vertical sections of sets and functions; it shows that sections of product-measurable sets and functions remain measurable, that the two section-measure integrals of a product-measurable set agree for sigma-finite factors, and constructs the product measure with its rectangle formula, sigma-finiteness, and uniqueness. Tonelli's theorem then identifies the double integral of a nonnegative measurable function on a sigma-finite product with the two iterated integrals, and Fubini's theorem extends this to L1 functions, showing that almost every section is integrable and the three integrals coincide. Variants for the completed product relax section measurability to hold only almost everywhere. Euclidean consistency is established: Borel products of Euclidean spaces are Euclidean Borel, the product of Lebesgue measures agrees with Lebesgue measure on Borel sets, and Euclidean Lebesgue measure is the completion of that product. The layer-cake formula computes the integral of a power from the distribution function, and polar-coordinate and C1 change-of-variables formulas are available. On the signed side, the section defines signed and complex measures, positive, negative, and null sets, total variation, mutual singularity, and absolute continuity; it proves the Hahn and Jordan decompositions, the finiteness of complex-measure total variation, and the fact that total variation is itself a measure. Integration against signed and complex measures is developed through simple integrals, dominated simple approximations, and the variation bound. Lebesgue decomposition and the Radon–Nikodym theorem supply densities unique up to almost-everywhere equality, together with additivity, chain-rule, epsilon-delta, and polar-decomposition results. A final strand treats absolute continuity of functions on a compact interval: derivatives exist almost everywhere, the sharp fundamental theorem of calculus characterizes absolutely continuous functions, and integration by parts and change of variables are derived. Standard probability tools such as uniform integrability and the monotone class theorem have no dedicated library identifiers here and are not assumed.

Landmark items: the iterated-section-measure agreement for sigma-finite measures; existence, rectangle formula, and uniqueness of the sigma-finite product measure; Tonelli's theorem for nonnegative functions; Fubini's theorem for L1 functions; the Hahn decomposition; the Jordan decomposition; finiteness of complex total variation; the Lebesgue decomposition for sigma-finite signed measures; the Radon–Nikodym density existence and uniqueness; the chain rule for Radon–Nikodym derivatives; the polar decomposition of finite signed and complex measures; the Euclidean Lebesgue measure as the completion of the product; the layer-cake formula; the polar-coordinates formula; the fundamental theorem of calculus for absolutely continuous functions; integration by parts; change of variables for increasing absolutely continuous maps; the Banach–Zarecki characterization; and constancy of absolutely continuous functions with zero derivative almost everywhere.

This section supplies the measure-theoretic differentiation and distribution-function machinery needed before a rigorous introduction to probability. It works through three library reading pages in order. The first page, Differentiation of Monotone Functions and the Vitali Covering Theorem, develops Dini derivatives, fine and Vitali covers on the line, Riesz's rising-sun lemma, the one-sided Hardy–Littlewood inequalities, and the almost-everywhere differentiability of monotone functions, together with the jump/continuous split and Fubini's term-by-term differentiation for series of increasing functions. The second page, The Maximal Function and Lebesgue Differentiation, moves to R^n: locally integrable functions, ball averages, centered and uncentered maximal functions, weak and strong type (p,q) operators, Lebesgue points, density, families shrinking nicely, the fivefold-dilate Vitali lemma, the weak (1,1) maximal inequality, Marcinkiewicz interpolation, the Lebesgue differentiation theorem, the density theorem, differentiation of Borel measures finite on compact sets, and the L^1 fundamental theorem of calculus. The third page, Lebesgue–Stieltjes Measures and Distribution Functions, constructs the half-open interval algebra, the Stieltjes premeasure, the Lebesgue–Stieltjes measure, uniqueness, normalized distribution functions, interval formulas and atoms, regularity, the Cantor measure, and the atomic-plus-atomless decomposition of finite Borel measures. Missing prerequisites assumed from earlier sections are sigma-algebras, measures, outer measure and Carathéodory extension, measurable functions, the Lebesgue integral with its convergence theorems, L^p spaces, and the Radon–Nikodym and Lebesgue decomposition theorems; these are not re-derived here. No dedicated library page on the monotone class or pi-lambda theorem, and none explicitly on uniform integrability, appears in the supplied catalog; the reader should note these as library gaps when proceeding to probability.

Landmark results to anchor the reading: Dini derivatives always exist in the extended reals and detect finite differentiability; Riesz's rising-sun lemma with its endpoint conclusion; the Vitali covering theorem for fine covers on the line; the monotonicity of monotone functions' almost-everywhere differentiability by the rising-sun and Lebesgue–Stieltjes routes; the derivative of a nondecreasing function is measurable, integrable, and bounded by its total increase; the unique absolutely-continuous plus jump plus singular-continuous decomposition; bounded-variation functions are differentiable almost everywhere; the fivefold-dilate Vitali covering lemma for balls; the centered Hardy–Littlewood maximal operator is weak type (1,1); Marcinkiewicz interpolation; the Lebesgue differentiation theorem on R^n; almost every point is a Lebesgue point; the Lebesgue density theorem; differentiation along families shrinking nicely; differentiation of sigma-finite Borel measures finite on compact sets; the L^1 fundamental theorem of calculus; existence, uniqueness, and the correspondence theorem for Lebesgue–Stieltjes measures; and the atomic-plus-atomless decomposition of finite Borel measures.

This section supplies the two measure-theoretic pillars that a rigorous introduction to probability leans on, and it presupposes only the real-analysis material already accumulated: limits of sequences and functions, continuity, differentiability, and the earlier sections of this course on measure spaces, integration, Lp spaces, and product measures. The first pillar is the Riesz–Markov–Kakutani theory: positive linear functionals on C_c(X) over a locally compact Hausdorff space are represented by integration against a Radon measure, and the representing measure is unique. The building blocks are the definitions of C_c and C_0, of the cutoff relation f≺U, of positive functionals, and of Radon, regular, and regular complex Borel measures. The construction proceeds through the functional outer content, Carathéodory measurability of open sets, inner regularity on opens, and the representation and uniqueness theorems.

Companion results identify when locally finite Borel measures are regular and record that Lebesgue measure on R^n is Radon, with applications to Lusin's theorem, density of C_c in Lp, and the dual of C_0. The second pillar develops density, separability, and convolution in Lp: simple functions, box steps, and C_c (then C_c^∞) are dense, rational box steps give a countable dense set, countably generated sigma-finite spaces yield separable Lp while L^∞[0,1] is not separable, translation is continuous in Lp, and convolution is treated through existence and the L1 bound, Young's and Minkowski's integral inequalities, smoothness of mollifier convolutions, and convergence of approximate identities in Lp. Library gaps: the catalog has no dedicated page on monotone-class/Dynkin arguments, on uniform integrability as a standalone topic, or on pre-measure-theoretic elementary probability; these would need separate sourcing before the probability sections. No page on Lebesgue measure of the line alone is present, though Euclidean-space material covers it.

This section builds the measure-theoretic probability machinery needed for a rigorous introduction to probability theory, starting from the supplied real-analysis prerequisites (limits of sequences, limits of functions, continuity and differentiability) and the preceding measure, integration, Lp, product-measure, and distribution-function material. The three reading pages develop probability spaces, random variables and expectation; independence, Borel–Cantelli and zero-one laws; and the modes of convergence for random variables.

Landmark results anchor each stage: the correspondence between laws and distribution functions, the change-of-variables formula for expectation, Jensen's inequality, the normal equations for best affine L2 prediction, factorization of expectations over independent variables, and the relation between convergence notions. Library gaps are disclosed in the plan: there is no standalone Lebesgue-measure-on-R page, no dedicated monotone-class/Dynkin theorem item, no standalone uniform-integrability page, and advanced Borel–Cantelli and disintegration material is absent; these are not assumed and are not invented.

This section assembles, from the six listed reading pages, the measure-theoretic machinery that carries the reader from real-analysis limits, continuity and differentiability to a rigorous introduction to probability. It has two arcs. The first constructs probability measures on infinite products: finite-coordinate cylinders, their sigma-algebra, consistent finite-dimensional laws, and Kolmogorov extension for standard-Borel coordinates, so that a stochastic process is pinned down by its finite-dimensional distributions. The second arc develops limit theorems on the resulting spaces: weak laws via Chebyshev and truncation for independent arrays, strong laws via the Kolmogorov maximal inequality, the convergence criterion and three-series theorem, central limit theorems from Lindeberg-Levy through Lindeberg-Feller and the multivariate case, characteristic functions with inversion and the Levy continuity theorem, and weak convergence, tightness, Prokhorov and Skorokhod representation on Polish spaces.

Library gaps are disclosed: the catalog has no dedicated monotone-class/pi-lambda page, no standalone uniform-integrability page, no pre-measure-theoretic discrete-probability page, and treatment of characteristic functions versus weak convergence is distributed across two pages rather than one. Reachability of the stated real-analysis prerequisites from the assumed pages is not verified by this library section, so those prerequisite pages must be read first. Landmark definitions and theorems (non-exhaustive): Kolmogorov extension for standard-Borel coordinates; countable products of probability spaces; process laws determined by finite-dimensional distributions; Kolmogorov maximal inequality and convergence criterion; Kolmogorov three-series theorem; Kolmogorov IID L1 strong law and its necessity of integrability; Lindeberg-Levy and Lindeberg-Feller central limit theorems with the Feller converse; uniqueness of a law from its characteristic function, Levy inversion and the Levy continuity theorem; the Cramer-Wold device; the Portmanteau theorem; Prokhorov tightness theorem; Skorokhod representation; Levy-Prokhorov metric metrizing weak convergence; plus the constituent definitions of cylinders, consistency, IID, truncation, tightness and characteristic functions.

This section builds the measure-theoretic machinery for a rigorous introduction to probability, starting from real analysis (limits, continuity, differentiability) and the preceding measure-theory sections. It opens with conditional expectation, defined via the Radon–Nikodym theorem, then develops its algebra, order, convergence, Jensen, and L2-projection properties, together with uniform integrability of conditional expectations. Conditional distributions and regular conditional probability follow, using measure and probability kernels, kernel integration measurability, disintegration of joint laws on standard Borel spaces, conditional density formulas, and Bayes' formula for dominated kernels. Discrete-time martingales are then introduced through filtrations, adapted integrable processes, natural filtrations, martingale differences, convex-function submartingales, predictable processes, martingale transforms, Doob decomposition, and predictable quadratic variation.

Martingale inequalities and convergence cover upcrossing numbers, Doob's submartingale convergence and L1/Lp maximal inequalities, Lp-bounded and uniformly integrable convergence, the closed martingale characterization, reverse martingales, Lévy upward and downward convergence, Azuma–Hoeffding, and the martingale central limit theorem. Stopping times conclude the section with the sigma-algebra at a stopping time, stopped processes, the stopped-martingale theorem, optional sampling for bounded times, and optional stopping under uniform integrability, integrable time with bounded increments, and a dominating integrable variable. Library gaps remain: no dedicated Lebesgue-measure-on-R page, no standalone monotone class or Dynkin pi-lambda page, no standalone uniform integrability page, and limited disintegration and conditional Borel–Cantelli material, so those prerequisites are not assumed as supplied.