An Introduction to Measure Theory — Chapter-by-Chapter Outline

Author: Terence Tao First published: 2011 Edition covered: First edition, Graduate Studies in Mathematics, vol. 126, American Mathematical Society. 206 pp. (The book derives from lecture notes posted on Tao's blog terrytao.wordpress.com; a freely available PDF matches the published text.)

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Central thesis

Measure theory solves a fundamental problem: given an arbitrary subset E of Euclidean space ℝᵈ, what is its "size"? The naive idea — that size equals the sum of point sizes — immediately breaks down (the product ∞ · 0 is indeterminate), and finite-partition arguments run into the Banach–Tarski paradox. The resolution is to restrict attention to a large but not universal class of measurable sets and assign them a measure that is countably additive. This one design choice — countable rather than merely finite additivity — is what separates Lebesgue theory from the older Jordan–Riemann theory and is responsible for its superior behavior under limits.

The book argues that Lebesgue measure and the Lebesgue integral are best understood not as isolated constructions but as a completion of the classical Riemann theory: every Jordan-measurable set is Lebesgue measurable, every Riemann-integrable function is Lebesgue integrable, and the Lebesgue theory extends both while gaining the crucial property that limits of measurable sets and functions remain measurable. The payoff appears in three celebrated convergence theorems (monotone convergence, Fatou's lemma, dominated convergence) that fail for Riemann integrals.

The text then lifts the entire theory from Euclidean space to abstract measure spaces (X, B, μ) — triples of a set, a σ-algebra, and a countably additive measure — and shows that integration, convergence, and differentiation all have purely measure-theoretic formulations that unify probability theory, harmonic analysis, and ergodic theory.

How does one assign a consistent, countably additive notion of size to subsets of ℝᵈ that is preserved under the passage to limits?

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Chapter 1 — Measure theory

Central question

How are the various components of measure and integration theory — elementary measure, Lebesgue measure, the Lebesgue integral, abstract measure spaces, modes of convergence, differentiation theorems, and outer-measure constructions — built up from scratch and connected to each other?

Main argument

This single large chapter (pages 1–207) is divided into seven sections that form the core of the book. Each section is treated below as a named sub-chapter.

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Section 1.1 — Prologue: The problem of measure

Central question

Why is it hard to assign a notion of "size" to an arbitrary subset of ℝᵈ, and what are the two classical answers — Jordan measure and the Riemann integral — before the Lebesgue theory is introduced?

Main argument

The problem stated. Tao opens by asking for a function m(E) that assigns a non-negative real number (or +∞) to every subset E ⊂ ℝᵈ, extending the intuitive notions of length, area, and volume. Two natural desiderata immediately conflict: (i) m should be translation-invariant and agree with elementary geometry on boxes; (ii) m should be defined for all subsets. The Banach–Tarski paradox — which asserts that a unit ball in ℝ³ can be decomposed into five pieces and reassembled into two disjoint copies of the ball — shows these desiderata are logically incompatible. The standard resolution is to abandon the goal of measuring every subset and instead identify a large class of measurable sets that behaves well.

Elementary measure. The simplest measurable sets are elementary sets: finite unions of boxes (Cartesian products of intervals). The elementary measure m(E) is defined as the total volume of any disjoint box decomposition, and Tao proves this is well-defined (independent of decomposition) via a discretization argument involving the lattice (1/N)ℤᵈ. Elementary measure is finitely additive but not countably additive.

Jordan measure. Jordan measure extends elementary measure by an approximation procedure. The Jordan outer measure m∗,(J)(E) is the infimum of elementary measures of elementary supersets of E; the Jordan inner measure m∗,(J)(E) is the supremum of elementary measures of elementary subsets. A set E is Jordan measurable if these agree. Jordan measurable sets form a Boolean algebra closed under finite unions and intersections. The Riemann integral of a bounded function on [a,b] is then characterized as the unique linear functional on bounded Riemann-integrable functions satisfying linearity, monotonicity, and the property that ∫1_E = m(E) for Jordan measurable E.

Limitations of Jordan measure. The rationals ℚ ∩ [0,1] have Jordan outer measure 1 but Jordan inner measure 0, so they are not Jordan measurable. Countable unions of Jordan measurable sets need not be Jordan measurable. Pointwise limits of Riemann-integrable functions need not be Riemann-integrable (even when uniformly bounded). These failures motivate the Lebesgue theory of the next section.

Key ideas

• An elementary set is a finite union of boxes; its measure is the sum of box volumes, independent of how the boxes are chosen.
• Jordan outer measure: infimum over elementary supersets; Jordan inner measure: supremum over elementary subsets.
• A set is Jordan measurable iff its boundary has Jordan outer measure zero.
• The Banach–Tarski paradox establishes that no translation-invariant, finitely additive measure can be defined on all subsets of ℝ³; the proof requires the axiom of choice.
• The problem of measure splits into: (i) which sets are measurable? (ii) how is their measure defined? (iii) what axioms does measure obey? (iv) are ordinary geometric sets measurable? (v) does measure agree with naive geometry on ordinary sets?
• The Riemann integral corresponds exactly to Jordan measure: f is Riemann integrable on [a,b] iff the region under its graph is Jordan measurable.
• Jordan measure is only finitely subadditive; Lebesgue theory requires countable subadditivity.

Key takeaway

The classical Jordan–Riemann theory provides an elementary but limited measure theory; its failure to handle countable unions and pointwise limits drives the construction of Lebesgue measure.

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Section 1.2 — Lebesgue measure

Central question

How does replacing finite coverings with countable coverings in the definition of outer measure produce a dramatically more powerful theory, and what does the resulting class of Lebesgue measurable sets look like?

Main argument

Lebesgue outer measure. The key modification is to define the Lebesgue outer measure m∗(E) as the infimum of total volumes ∑|Bₙ| over all countable covers of E by boxes Bₙ:

$$m^*(E) := \inf{\bigcup{n=1}^\infty Bn \supset E} \sum{n=1}^\infty |B_n|$$

This immediately gives a much smaller outer measure than the Jordan version: every countable set — including the rationals — has Lebesgue outer measure zero (the ε/2ⁿ trick: cover the nth point by a box of side ε/2ⁿ, giving total volume Cₐεᵈ → 0). Lebesgue outer measure satisfies three axioms: m∗(∅) = 0, monotonicity (E ⊂ F ⟹ m∗(E) ≤ m∗(F)), and countable subadditivity m∗(⋃Eₙ) ≤ ∑m∗(Eₙ).

Lebesgue measurability. A set E ⊂ ℝᵈ is Lebesgue measurable if for every ε > 0 there exists an open set U ⊃ E with m∗(U\E) ≤ ε. When E is measurable, its Lebesgue measure is m(E) := m∗(E). Tao notes this captures Littlewood's first principle: measurable sets are "almost open." Equivalent characterizations include the Carathéodory criterion (m∗(A) = m∗(A∩E) + m∗(A\E) for all A), approximation from above by Gδ sets, and approximation from below by Fσ sets.

Properties of Lebesgue measure. The Lebesgue measurable sets form a σ-algebra L[ℝᵈ]: closed under complements and countable unions. Lebesgue measure is countably additive on disjoint measurable sets. It extends Jordan measure: every Jordan measurable set is Lebesgue measurable, and the two measures agree. Translation invariance and uniqueness (Lebesgue measure is the unique translation-invariant Borel measure assigning measure 1 to the unit cube, up to normalization) are established.

Non-measurable sets. Using the axiom of choice one can construct non-measurable sets (Vitali's construction). This shows that the σ-algebra cannot be all of 2^ℝᵈ. In practice, sets arising in analysis are always measurable; the exceptions are pathological axiom-of-choice constructions.

Key ideas

• The ε/2ⁿ trick: to show that countably many points have zero outer measure, cover the nth point by a box of side proportional to ε/2ⁿ.
• Lebesgue outer measure is countably subadditive; Jordan outer measure is only finitely subadditive — this gap is fundamental.
• Lebesgue measurability defined via open approximation: E is measurable iff for every ε > 0 an open set U ⊃ E exists with m∗(U\E) ≤ ε.
• The σ-algebra L[ℝᵈ] includes all open sets, all closed sets, all Gδ and Fσ sets, and all sets of measure zero (null sets).
• Countable additivity: for disjoint measurable Eₙ, m(⋃Eₙ) = ∑m(Eₙ).
• Upward/downward monotone convergence for sets: if E₁ ⊂ E₂ ⊂ … then m(⋃Eₙ) = lim m(Eₙ); downward holds when some m(Eₙ) < ∞.
• Non-measurable sets exist (Vitali sets via axiom of choice) but do not arise in ordinary analysis.

Key takeaway

Replacing finite coverings by countable ones converts subadditivity into the crucial tool of countable additivity, yielding a measure theory that handles limits.

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Section 1.3 — The Lebesgue integral

Central question

How does one extend the classical Riemann integral to a Lebesgue integral that handles a vastly larger class of functions and obeys good convergence theorems?

Main argument

Two parallel theories. Tao builds two integrals in parallel, mirroring the structure of Lebesgue measure. The unsigned integral ∫f dx for f : ℝᵈ → [0,+∞] is defined first; the absolutely integrable integral ∫f dx for f : ℝᵈ → ℂ is defined in terms of it. This mirrors the construction of unsigned vs. absolutely convergent infinite series.

Simple functions. A simple function is a finite linear combination f = c₁1{E₁} + … + cₖ1{Eₖ} of indicator functions of measurable sets. The integral of a simple function is defined naturally: ∫f = c₁m(E₁) + … + cₖm(Eₖ). This is well-defined (independent of the representation) by the properties of Lebesgue measure.

Unsigned integral. For a general unsigned measurable function f : ℝᵈ → [0,+∞], the unsigned Lebesgue integral is:

$$\int{\mathbb{R}^d} f(x)\,dx := \sup{0 \leq g \leq f,\, g \text{ simple}} \int g(x)\,dx$$

This is a "lower integral" construction — analogous to the lower Darboux integral — and always exists in [0,+∞]. Key properties: monotonicity, linearity (Theorem 1.3.15 and its proof via finite additivity + limiting), and Markov's inequality P(f ≥ λ) ≤ (1/λ)∫f.

Absolutely integrable integral. f : ℝᵈ → ℂ is absolutely integrable if ∫|f| dx < ∞. One then decomposes f into positive/negative (or real/imaginary) parts and integrates each. The space of absolutely integrable functions is denoted L¹(ℝᵈ); modulo the equivalence relation "equal almost everywhere" it becomes a normed space with ‖f‖_{L¹} = ∫|f| dx.

Littlewood's three principles. Tao collects three principles that capture the essence of the Lebesgue theory:

1. Every measurable set is "almost" a finite union of intervals (Littlewood's first principle — formalizes the open-approximation definition).
2. Every measurable function is "almost" continuous — Lusin's theorem: for every measurable f and ε > 0, there exists a closed set F with m(ℝᵈ\F) ≤ ε such that f|_F is continuous.
3. Every convergent sequence of measurable functions "almost" converges uniformly — Egorov's theorem: if fₙ → f pointwise a.e. then for every ε > 0 there is a measurable set A with m(A) ≤ ε such that fₙ → f uniformly outside A.

Density and approximation. Every absolutely integrable function can be approximated in L¹ norm by simple functions, by step functions, or by continuous compactly supported functions. This density result is proved by a "three-step" argument: simple → step → continuous.

Key ideas

• The unsigned integral is defined via a supremum over simple lower bounds, exactly paralleling the lower Darboux integral.
• The integral of a non-negative simple function f = ∑cₙ1_{Eₙ} is ∑cₙm(Eₙ), requiring careful handling of the 0 · ∞ = 0 convention.
• The Lebesgue integral extends the Riemann integral: every Riemann-integrable function is Lebesgue-integrable with the same value.
• Lusin's theorem: measurable functions are almost continuous (delete a set of small measure).
• Egorov's theorem: pointwise a.e. convergence is almost uniform convergence.
• The Lebesgue and Riemann integrals are related by: f is Riemann-integrable on [a,b] iff f is bounded and measurable, and discontinuous only on a set of measure zero.
• Lebesgue measure and integral are completions of Jordan measure and the Riemann integral, in the same sense that ℝ is the completion of ℚ.

Key takeaway

The Lebesgue integral, built via simple functions and a supremum construction, extends Riemann integration while gaining the compactness properties expressed by Lusin's and Egorov's theorems.

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Section 1.4 — Abstract measure spaces

Central question

What is the minimal abstract structure needed to reproduce all the key results of Lebesgue theory, and what are the fundamental convergence theorems in this abstract setting?

Main argument

Measure spaces. A measure space is a triple (X, B, μ) where X is a set, B is a σ-algebra (a collection of subsets closed under complements and countable unions), and μ : B → [0,+∞] is a countably additive measure (μ(∅) = 0 and μ(⋃Eₙ) = ∑μ(Eₙ) for disjoint Eₙ). A measurable space (X, B) has no measure specified. Lebesgue measure on ℝᵈ is the leading example. Other examples include counting measure, Dirac measures δₓ, and Gaussian measures on ℝⁿ.

σ-algebras. Tao discusses how σ-algebras arise: the Borel σ-algebra B[X] is the smallest σ-algebra containing all open sets of a topological space X. It is generated by the open sets and contains all open, closed, Gδ, Fσ sets. The σ-algebra generated by a collection A is denoted ⟨A⟩.

Integration on abstract measure spaces. Simple functions and their integrals are defined exactly as in the Euclidean case, but now indicator functions 1_E have integral μ(E). The unsigned integral and absolutely convergent integral are constructed by the same supremum/decomposition procedure. Key properties — linearity, monotonicity, Markov's inequality — all carry over.

The three convergence theorems. These are the payoff of the abstract setup:

• Monotone convergence theorem (MCT, Theorem 1.4.44): if 0 ≤ f₁ ≤ f₂ ≤ … are unsigned measurable functions with fₙ → f pointwise, then ∫fₙ dμ → ∫f dμ. Tao's proof approximates each fₙ by a simple function and uses the increasing structure to pass to the limit.

• Fatou's lemma (Corollary 1.4.47): for unsigned measurable fₙ ≥ 0, $$\int \liminf{n\to\infty} fn\, d\mu \leq \liminf{n\to\infty} \int fn\, d\mu.$$ Informally: mass can be destroyed in the limit (by the "escape to infinity"), but cannot be created. Tao gives three canonical "escape to infinity" examples on ℝ: escape to horizontal infinity (fn = 1{[n,n+1]}), escape to width infinity (fn = (1/n)1{[0,n]}), and escape to vertical infinity (fn = n·1{[1/n,2/n]}).

• Dominated convergence theorem (DCT, Theorem 1.4.49): if fₙ → f pointwise a.e. and |fₙ| ≤ G a.e. for some absolutely integrable G, then ∫fₙ dμ → ∫f dμ. Proof: apply Fatou's lemma to fₙ + G and G − fₙ simultaneously.

Completeness, atoms, and Lp. A measure space is complete if every subset of a null set is measurable. The completion of any measure space is defined. The L¹ seminorm ‖f‖_{L¹} = ∫|f| dμ and convergence in L¹ are introduced.

Key ideas

• A σ-algebra is stronger than a Boolean algebra: it is closed under countable (not just finite) unions and intersections.
• The distinction between σ-finite and non-σ-finite measure spaces: many theorems (Fubini, uniqueness of extensions) require σ-finiteness.
• Monotone convergence theorem: the fundamental tool; proved by an explicit approximation of unsigned measurable functions by simple ones.
• Fatou's lemma: ∫lim inf fₙ ≤ lim inf ∫fₙ — the asymmetry between ∫lim inf and lim inf ∫ reflects the possibility that mass "escapes to infinity."
• Dominated convergence theorem: if a dominating integrable function G exists, limits and integrals commute; this is the tool that makes the Lebesgue integral practical.
• The Borel-Cantelli lemma: if ∑μ(Eₙ) < ∞, then almost every point lies in at most finitely many Eₙ. Proved via Tonelli's theorem for series.
• Tonelli's theorem for sums and integrals: ∫(∑fₙ) dμ = ∑(∫fₙ dμ) for unsigned fₙ (Corollary 1.4.46, proved from MCT).

Key takeaway

The three convergence theorems — monotone convergence, Fatou's lemma, and dominated convergence — are the engine of the Lebesgue theory and require only the abstract measure-space axioms, not the specific structure of ℝᵈ.

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Section 1.5 — Modes of convergence

Central question

In what different senses can a sequence of measurable functions converge, how do these modes relate to each other, and under what additional hypotheses do implications hold that fail in general?

Main argument

Seven modes of convergence. Once the domain X carries a measure, a sequence fₙ : X → ℂ can converge to f in any of the following ways, listed roughly from strongest to weakest:

1. Uniform convergence: sup_x |fₙ(x) − f(x)| → 0.
2. L∞ norm / essential uniform convergence: ‖fₙ − f‖_{L∞} → 0.
3. Almost uniform convergence: for every ε > 0 there is a set E with μ(E) ≤ ε outside which convergence is uniform.
4. Pointwise convergence (everywhere): fₙ(x) → f(x) for every x.
5. Pointwise almost everywhere (a.e.) convergence: fₙ(x) → f(x) for μ-a.e. x.
6. L¹ norm convergence: ‖fₙ − f‖_{L¹} → 0.
7. Convergence in measure: for every ε > 0, μ({x : |fₙ(x) − f(x)| ≥ ε}) → 0.

The diagram of implications. Uniform ⟹ L∞ ⟹ almost uniform ⟹ pointwise a.e. Uniform ⟹ pointwise everywhere ⟹ pointwise a.e. L¹ ⟹ in measure. Almost uniform ⟹ in measure. No other implications hold in general.

Four canonical counterexamples distinguish these modes on (ℝ, m):

• Escape to horizontal infinity: fₙ = 1_{[n,n+1]} converges pointwise to 0 but not in measure or L¹.
• Escape to width infinity: fₙ = (1/n)1_{[0,n]} converges uniformly to 0 but not in L¹.
• Escape to vertical infinity: fₙ = n·1_{[1/n,2/n]} converges pointwise and almost uniformly to 0 but not in L¹.
• Typewriter sequence: fₙ = 1_{[(n−2ᵏ)/2ᵏ, (n−2ᵏ+1)/2ᵏ]} (the indicator of a small interval marching repeatedly across [0,1]) converges in measure and L¹ but not pointwise a.e. This is the crucial example showing that L¹ convergence does not imply pointwise a.e. convergence.

Recovery of implications under additional hypotheses. On a finite measure space, almost uniform convergence is equivalent to pointwise a.e. convergence (Egorov's theorem). L¹ convergence implies in-measure convergence always; the converse requires uniform integrability.

Uniform integrability. A sequence fₙ is uniformly integrable if (i) supₙ‖fₙ‖{L¹} < ∞, (ii) ∫{|fₙ|≥M}|fₙ| dμ → 0 as M → ∞ (no escape to vertical infinity), and (iii) ∫_{|fₙ|≤δ}|fₙ| dμ → 0 as δ → 0 (no escape to width infinity). Theorem 1.5.13: a uniformly integrable sequence converges in L¹ iff it converges in measure. Every dominated sequence is uniformly integrable.

Key ideas

• No single mode of convergence dominates all others; the seven modes form a partial order, not a chain.
• The three "escape to infinity" paradigms (horizontal, width, vertical) each exemplify a different failure mode.
• The typewriter sequence is the key example showing that L¹ convergence (or convergence in measure) does not guarantee pointwise a.e. convergence.
• Egorov's theorem recovers almost uniform convergence from pointwise a.e. convergence on sets of finite measure.
• Uniform integrability is the natural condition that rules out all three escapes to infinity simultaneously and allows equivalence of L¹ convergence and convergence in measure.
• Convergence in L¹ norm implies convergence of integrals (by the triangle inequality); no other mode automatically does.

Key takeaway

Different modes of convergence capture different failure mechanisms (escape to horizontal/vertical/width infinity and the typewriter phenomenon), and uniform integrability is the key condition that brings them into alignment.

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Section 1.6 — Differentiation theorems

Central question

Under what conditions can one differentiate a function (or an integral) almost everywhere, and how do differentiation and integration serve as approximate inverses of each other in the Lebesgue setting?

Main argument

Fundamental theorem of calculus, version 1. If f is absolutely integrable on ℝ, its indefinite integral F(x) = ∫_{[−∞,x]} f(t) dt is of bounded variation and is differentiable almost everywhere with F′(x) = f(x) a.e. This is the Lebesgue differentiation theorem (Theorem 1.6.12).

The Lebesgue differentiation theorem. For any locally absolutely integrable f : ℝ → ℂ and almost every x,

$$\lim{h \to 0^+} \frac{1}{h}\intx^{x+h} f(t)\, dt = f(x).$$

The proof uses the density argument: (a) establish the result for the dense subclass of continuous functions (where it is trivial), then (b) apply the Hardy–Littlewood maximal inequality to propagate the result to all absolutely integrable functions.

Hardy–Littlewood maximal inequality. The one-sided Hardy–Littlewood maximal function is f∗(x) = sup{h>0} (1/h)∫{[x,x+h]}|f|. The maximal inequality (Lemma 1.6.16) states:

$$m({x : f^*(x) \geq \lambda}) \leq \frac{1}{\lambda}\int_\mathbb{R}|f|\, dt.$$

The proof uses the rising sun lemma: if the sun shines horizontally from +∞ onto the graph of the function F(x) = ∫_{[a,x]}|f(t)| dt − (x−a)λ, the set of points in shadow is a countable union of open intervals In = (aₙ,bₙ) such that F(bₙ) ≥ F(aₙ). Summing lengths: ∑(bₙ − aₙ) ≤ (1/λ)∫|f|.

Monotone functions are differentiable a.e. Theorem 1.6.25: every monotone non-decreasing function F : [a,b] → ℝ is differentiable almost everywhere. The proof is built in stages:

• For continuous monotone functions, the density argument and the Hardy–Littlewood inequality give the result.
• For jump functions F = ∑cₙJₙ (countable superpositions of basic unit-step functions Jₙ), the same density argument applies using piecewise-constant jump functions as the dense subclass.
• The general case follows from the continuous-singular decomposition (Lemma 1.6.31): every bounded monotone non-decreasing function decomposes uniquely as F = Fc + Fpp where Fc is continuous monotone and Fpp is a jump function.

Bounded variation and absolute continuity. A function F has bounded variation if ‖F‖_{TV} = sup ∑|F(xᵢ₊₁) − F(xᵢ)| < ∞. Bounded variation functions decompose as differences of monotone functions (Proposition 1.6.34) and are therefore differentiable a.e. (BV differentiation theorem, Corollary 1.6.35). A function F is absolutely continuous on [a,b] if for every ε > 0 there exists δ > 0 such that ∑|F(bₖ) − F(aₖ)| ≤ ε whenever the intervals (aₖ,bₖ) are disjoint with ∑(bₖ − aₖ) ≤ δ. Absolute continuity is the exact condition needed for the second fundamental theorem of calculus: F(b) − F(a) = ∫ₐᵇ F′(t) dt.

Key ideas

• The Lebesgue differentiation theorem says that almost every point x is a Lebesgue point: the average of f over [x, x+h] converges to f(x) as h → 0.
• The density argument pattern: prove for a dense subclass + establish a quantitative maximal inequality ⟹ extend to the whole class.
• The Hardy–Littlewood maximal inequality is a weak-type (1,1) inequality; it is the foundational inequality of harmonic analysis.
• The rising sun lemma gives a soft-analysis proof of the maximal inequality by a greedy geometric covering argument.
• The Dini derivatives D⁺F, D⁻F, D₊F, D₋F are four one-sided limsup/liminf quotients; F is differentiable at x iff all four agree and are finite.
• Bounded variation ⟺ difference of two bounded monotone functions ⟺ differentiable a.e.
• Absolute continuity is strictly stronger than bounded variation and is the exact condition for the second fundamental theorem of calculus.
• The Rademacher differentiation theorem (Section 2.2) extends the Lipschitz a.e. differentiability from 1D to Rd using Fubini's theorem.

Key takeaway

Monotone, bounded-variation, and absolutely continuous functions are all differentiable almost everywhere, and absolute continuity characterizes exactly when the fundamental theorem of calculus holds in the Lebesgue setting.

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Section 1.7 — Outer measures, pre-measures, and product measures

Central question

How can one construct new measures — Lebesgue-Stieltjes measures, product measures — using a general machine, and what theorem allows integration over a product space to be computed by iterated integration?

Main argument

Abstract outer measures. An outer measure μ∗ : 2^X → [0,+∞] is any function satisfying μ∗(∅)=0, monotonicity, and countable subadditivity. The Lebesgue outer measure m∗ on ℝᵈ is the prototype.

Carathéodory's extension theorem. Given an outer measure μ∗, a set E is Carathéodory measurable (Definition 1.7.2) if:

$$\mu^(A) = \mu^(A \cap E) + \mu^*(A \setminus E) \quad \text{for all } A \subset X.$$

Theorem 1.7.3: the Carathéodory-measurable sets form a σ-algebra B, and μ∗ restricted to B is a complete measure. This theorem abstracts the construction of Lebesgue measure from Lebesgue outer measure.

Pre-measures and the Hahn–Kolmogorov extension. A pre-measure μ₀ : B₀ → [0,+∞] on a Boolean algebra B₀ satisfies μ₀(∅)=0 and countable additivity on disjoint unions contained in B₀. Theorem 1.7.8 (Hahn–Kolmogorov): every pre-measure extends to a countably additive measure on the σ-algebra ⟨B₀⟩ generated by B₀. Under σ-finiteness, the extension is unique. This is the abstract machine behind Lebesgue measure (extend elementary measure from the Boolean algebra of elementary sets).

Lebesgue-Stieltjes measure. Given any monotone non-decreasing F : ℝ → ℝ, there is a unique Borel measure μF (the Lebesgue-Stieltjes measure) such that μF([a,b]) = F₊(b) − F₋(a) and so on for all interval types (Theorem 1.7.9). This measure generalizes Lebesgue measure (take F(x) = x) and provides the foundation for integration with respect to arbitrary distribution functions in probability theory.

Product measures. Given σ-finite measure spaces (X, BX, μX) and (Y, BY, μY), their product measure μX × μY is the unique measure on the product σ-algebra BX × BY satisfying (μX × μY)(E × F) = μX(E)·μY(F) for all measurable rectangles E × F (Proposition 1.7.11). The proof uses the Hahn–Kolmogorov theorem.

Tonelli's theorem (Theorem 1.7.15/1.7.18): for unsigned measurable f : X×Y → [0,+∞],

$$\int{X\times Y} f\, d(\muX \times \muY) = \intX \left(\intY f(x,y)\,d\muY(y)\right)d\muX(x) = \intY \left(\intX f(x,y)\,d\muX(x)\right)d\mu_Y(y).$$

Fubini's theorem (Corollary 1.7.23): for absolutely integrable f, the same equality of iterated integrals holds. The key tool in the proof is the monotone class lemma (Lemma 1.7.14): the smallest monotone class containing a Boolean algebra A equals ⟨A⟩. This allows the theorem to be verified first on indicator functions of product sets, then extended to all simple functions, and finally to general measurable functions via MCT.

Key ideas

• The Carathéodory criterion μ∗(A) = μ∗(A∩E) + μ∗(A\E) for all test sets A is the abstract definition of measurability in terms of an outer measure.
• A pre-measure is the right notion of "proto-measure" on a Boolean algebra; the Hahn–Kolmogorov theorem extends it to a full measure on the generated σ-algebra.
• σ-finiteness is essential for uniqueness of the extension; without it, multiple extensions can exist.
• The Lebesgue-Stieltjes measure μ_F of a monotone function F generalizes Lebesgue measure and is the natural measure for the integral ∫f(x) dF(x) in Stieltjes notation.
• Product σ-algebra BX × BY is generated by measurable rectangles E × F.
• The monotone class lemma is the key structural tool allowing one to prove statements first for a simple generating class and then extend to the full σ-algebra.
• Tonelli's theorem (unsigned functions): iterated integration in any order equals the double integral.
• Fubini's theorem (signed/complex functions): same as Tonelli, but absolute integrability is required.

Key takeaway

The Hahn–Kolmogorov machine converts any pre-measure (including elementary measure and Lebesgue-Stieltjes measures) into a full countably additive measure, and the Fubini–Tonelli theorem enables the reduction of multidimensional integration to iterated one-dimensional integrals.

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Chapter 2 — Related articles

Central question

How do measure-theoretic tools connect to problem-solving heuristics, higher-dimensional differentiation, probability theory, and the rigorous foundations of infinite random processes?

Main argument

This chapter collects four supplementary topics that extend or apply the core Chapter 1 theory.

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Section 2.1 — Problem solving strategies

Central question

What general heuristics and specific strategies are most useful for solving problems in real analysis and measure theory?

Main argument

This section is deliberately informal — the only such section in the book — and synthesizes pedagogical wisdom from Tao's teaching of the course. It is organized as a numbered list of strategies with explanations and examples from the text.

The epsilon of room. Many arguments proceed by proving X ≤ Y + ε for all ε > 0 and then concluding X ≤ Y. This "epsilon of room" technique is ubiquitous in analysis and allows one to replace strict inequalities with non-strict ones in limiting arguments.

Decompose rough objects into smooth ones. To prove something about a measurable function, first prove it for continuous functions, then for step functions or simple functions, and finally extend by density (using the L¹ approximation theorems). This structure underlies the proof of the Lebesgue differentiation theorem.

Work locally before globally. Euclidean space has infinite measure; many estimates are easier on a compact ball B(0,R). One can often piece together local estimates using countable covers and countable additivity.

Be willing to throw away a null set. The "Lebesgue philosophy": null sets are irrelevant. If something holds outside a set of measure zero, it holds "almost everywhere," which is good enough for integration.

Zeno's trick: cut one epsilon into countably many. The identity ε = ε/2 + ε/4 + ε/8 + … justifies infinite sequences of approximations each costing ε/2ⁿ, for a total cost of ε. This is behind every ε/2ⁿ trick in the text.

Interchange sums/integrals when possible. The Fubini–Tonelli theorem and Tonelli's theorem for series both justify interchange; knowing when to apply them is a key skill.

Pass to subsequences. A sequence converging in L¹ or in measure has a subsequence converging pointwise a.e. This is a recurring technique for upgrading weak convergence to almost everywhere convergence.

Key ideas

• The "density argument" pattern: prove on a dense subclass + quantitative estimate ⟹ general result.
• Uncountable unions should be replaced by countable ones whenever possible (measurability is preserved by countable but not uncountable operations).
• The lim sup and lim inf are more flexible than limits for arguments where existence of the limit is not yet established (used in the BV differentiation theorem proof).
• Drawing pictures and attempting to construct counterexamples is productive even when the statement is true, because the failure of the counterexample reveals the mechanism of the proof.

Key takeaway

Measure-theoretic arguments follow recognizable structural templates; fluency with these patterns — epsilon of room, density arguments, Zeno's trick, passing to subsequences — is as important as knowing the theorems themselves.

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Section 2.2 — The Rademacher differentiation theorem

Central question

Does the Lipschitz a.e.-differentiability theorem extend from functions f : ℝ → ℝ to f : ℝᵈ → ℝ, and how does Fubini's theorem enable the reduction?

Main argument

The Rademacher theorem (Theorem 2.2.4) states: every Lipschitz continuous function f : ℝᵈ → ℝ is totally differentiable (not merely directionally differentiable) at almost every x₀ ∈ ℝᵈ.

Step 1: directional differentiability a.e. For any fixed direction v ∈ ℝᵈ, the function x ↦ f(x + tv) is a Lipschitz function of t ∈ ℝ (with the same Lipschitz constant). By the one-dimensional Lipschitz differentiation theorem (Exercise 1.6.41), it is differentiable in t for almost every starting point x. Applying Fubini's theorem: the directional derivative Dvf(x) exists for a.e. x ∈ ℝᵈ.

Step 2: from directional to total differentiability. Using an integration-by-parts formula:

$$\int{\mathbb{R}^d} Dv f(x) g(x)\,dx = \int{\mathbb{R}^d} f(x) D{-v}g(x)\,dx$$

(proved via dominated convergence and translation invariance), one shows that Dvf(x) = v · ∇f(x) for a.e. x, where ∇f = (∂f/∂x₁, …, ∂f/∂xₐ). Since this holds for all rational v ∈ ℚᵈ and Lipschitz functions are uniformly continuous, a compact approximation argument (using the total boundedness of the unit sphere) upgrades directional differentiability over all of ℚᵈ to total differentiability.

Key ideas

• Total differentiability at x₀ requires a single linear map ∇f(x₀) such that f(x₀+h) = f(x₀) + h·∇f(x₀) + o(|h|).
• Directional and partial differentiability do not imply total differentiability without continuity hypotheses (illustrated by f(x,y) = x₁x₂/(x₁²+x₂²) at the origin).
• Fubini's theorem reduces the multi-dimensional theorem to the one-dimensional one: differentiate along lines parallel to each coordinate axis.
• The integration-by-parts formula moves the differential operator from the rough function f to the smooth test function g.

Key takeaway

Fubini's theorem serves as a powerful "dimension-reduction" tool: the d-dimensional Rademacher theorem follows from the 1-dimensional Lipschitz differentiation theorem by integrating along lines.

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Section 2.3 — Probability spaces

Central question

How does the abstract measure-space framework specialize to provide rigorous foundations for probability theory?

Main argument

A probability space is a measure space (Ω, F, P) with P(Ω) = 1. The components carry probabilistic interpretations: Ω is the sample space (all possible outcomes), F is the event space (σ-algebra of observable events), and P(E) is the probability of event E. This is Kolmogorov's axiomatization of probability (1933).

Translation dictionary. Every measure-theoretic concept has a probabilistic counterpart: measurable function ↦ random variable, integral ↦ expectation E[X] = ∫X dP, "almost everywhere" ↦ "almost surely," the Markov inequality ↦ P(X ≥ λ) ≤ E[X]/λ, the Borel–Cantelli lemma ↦ if ∑P(Eₙ) < ∞ then P(infinitely many Eₙ occur) = 0.

Examples. Discrete uniform probability measures on finite sets; Gaussian measures on ℝⁿ with density (1/√(2π))e^{−x²/2} dx. The impossibility of a translation-invariant probability measure on ℤ or ℝ (Exercises 2.3.1, 2.3.2) connects to the earlier discussion of non-measurable sets.

Key ideas

• The three components Ω, F, P of a probability space correspond precisely to the set, σ-algebra, and measure of an abstract measure space.
• Independence: events E, F are independent if P(E ∩ F) = P(E)P(F); random variables X, Y are independent if σ(X) and σ(Y) are independent sub-σ-algebras.
• The expectation E[X] = ∫_Ω X dP is an absolutely convergent integral; standard measure-theoretic results (MCT, DCT) give the corresponding probabilistic theorems.

Key takeaway

Probability theory is a special case of abstract measure theory; Kolmogorov's axioms are simply the measure-space axioms applied to measures of total mass 1.

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Section 2.4 — Infinite product spaces and the Kolmogorov extension theorem

Central question

How does one construct a probability measure on an infinite (possibly uncountable) product space, and why does this matter for the rigorous foundations of stochastic processes?

Main argument

The problem. One wishes to construct a measure on the product space XA = ∏{α∈A} Xα for an arbitrary index set A (which may be uncountable). For each finite set B ⊂ A, one has a compatible "finite-dimensional" probability measure μB on XB = ∏{α∈B} Xα. The compatibility condition is: if C ⊂ B, then the pushforward of μB under the projection π{B←C} equals μC.

Kolmogorov extension theorem (Theorem 2.4.3): if each μB is an inner regular probability measure (on a metric space), then there exists a unique probability measure μA on BA such that (πB)∗μA = μ_B for all finite B ⊂ A.

Proof strategy. Define μ₀ on the Boolean algebra B₀ of "cylindrical sets" {E ∈ B₀ : E = πB⁻¹(EB)} by μ₀(E) := μB(EB). Show μ₀ is a pre-measure: the key step uses the Heine–Borel theorem (compactness) and a diagonalization argument to show that if μ₀(FN) ≥ ε > 0 for all N (where FN are decreasing cylindrical sets with empty intersection), a contradiction is obtained. Apply the Hahn–Kolmogorov extension theorem.

Applications. The theorem constructs:

• Bernoulli measure on {0,1}^ℕ (countably infinite coin flips).
• Product gaussian measure on ℝ^ℕ (countably infinite independent gaussians).
• The Wiener process (Brownian motion) on a space of continuous functions (with additional regularity work).

Key ideas

• A measurable event in the infinite product can depend on at most countably many coordinates (Exercise 2.4.1).
• Inner regularity (every measurable set's measure is approximated from inside by compact sets) is the key hypothesis that allows the Heine–Borel argument to work.
• The proof uses Hahn–Kolmogorov (from Section 1.7) plus compactness to show the cylindrical pre-measure is genuinely countably additive.
• Without inner regularity, the extension may fail or not be unique.

Key takeaway

The Kolmogorov extension theorem, proved by combining the Hahn–Kolmogorov theorem with compactness, provides the rigorous foundation for constructing probability measures on function spaces, enabling a mathematical theory of stochastic processes.

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The book's overall argument

1. Section 1.1 (Prologue: The problem of measure) — establishes that assigning a consistent notion of size to arbitrary subsets of ℝᵈ is impossible for all subsets simultaneously, and that the classical Jordan–Riemann theory, while adequate for smooth functions and simple sets, fails for countable unions and pointwise limits.

2. Section 1.2 (Lebesgue measure) — shows that replacing finite covers by countable covers in the definition of outer measure produces Lebesgue measure, a σ-algebra of measurable sets, and countable additivity, resolving the limitations of Jordan measure.

3. Section 1.3 (The Lebesgue integral) — constructs the Lebesgue integral via simple functions, establishes Littlewood's three principles (measurable sets ≈ open sets, measurable functions ≈ continuous functions, a.e. convergence ≈ uniform convergence), and proves that the Lebesgue integral is the completion of the Riemann integral.

4. Section 1.4 (Abstract measure spaces) — lifts the entire theory to the axiomatic setting of σ-algebras and countably additive measures, proving the three fundamental convergence theorems (MCT, Fatou, DCT) in full generality.

5. Section 1.5 (Modes of convergence) — catalogs seven distinct modes in which measurable functions can converge, shows by four canonical counterexamples that these modes are not linearly ordered, and identifies uniform integrability as the key condition that restores equivalences between L¹ convergence and convergence in measure.

6. Section 1.6 (Differentiation theorems) — establishes that monotone and bounded-variation functions are differentiable almost everywhere (via the Hardy–Littlewood maximal inequality and the rising sun lemma), and characterizes absolute continuity as the exact condition for the second fundamental theorem of calculus in the Lebesgue setting.

7. Section 1.7 (Outer measures, pre-measures, and product measures) — provides the abstract construction machine (Carathéodory extension + Hahn–Kolmogorov theorem) that generates all measures from simpler data, and proves the Fubini–Tonelli theorem enabling iterated integration.

8. Section 2.1 (Problem solving strategies) — draws out the meta-patterns that underlie all of Chapter 1: epsilon of room, density arguments, Zeno's trick, passage to subsequences, and the "Lebesgue philosophy" of working modulo null sets.

9. Section 2.2 (Rademacher differentiation theorem) — demonstrates the power of Fubini's theorem as a dimension-reduction tool by deriving the higher-dimensional Lipschitz differentiability theorem from the one-dimensional version.

10. Section 2.3 (Probability spaces) — shows that abstract measure spaces specialize to probability spaces, providing the foundation for Kolmogorov's axiomatization of probability.

11. Section 2.4 (Infinite product spaces and the Kolmogorov extension theorem) — completes the arc by extending product measures to infinite index sets, enabling the rigorous construction of stochastic processes and Brownian motion.

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Common misunderstandings

Misunderstanding: "Lebesgue measure is just another way to define the area or volume of ordinary sets."

Lebesgue measure does agree with the classical notion of length, area, and volume on ordinary geometric sets. Its significance is that it extends to a vastly larger σ-algebra of sets — including all open, closed, Gδ, and Fσ sets, and limits thereof — while preserving countable additivity. The key advance is not a new answer on old inputs but an answer on new inputs where the Jordan theory gives no answer.

Misunderstanding: "If fₙ → f in L¹, then fₙ(x) → f(x) for almost every x."

This is false: the typewriter sequence converges to 0 in L¹ (and in measure) but fails to converge pointwise anywhere on [0,1]. What is true is that every L¹-convergent sequence has a subsequence that converges pointwise a.e.

Misunderstanding: "The dominated convergence theorem says that if fₙ → f pointwise, then ∫fₙ → ∫f."

The DCT requires an absolutely integrable dominating function G with |fₙ| ≤ G a.e. Without this, even uniform pointwise convergence can fail to give convergence of integrals: fₙ(x) = (1/n)1_{[0,n]}(x) → 0 uniformly, but ∫fₙ = 1 ↛ 0.

Misunderstanding: "Monotone functions are differentiable everywhere."

Monotone functions are differentiable almost everywhere (Theorem 1.6.25), but not everywhere. The Cantor function is the standard counterexample: it is continuous, monotone non-decreasing, and has derivative 0 almost everywhere (on the complement of the Cantor set), yet it increases from 0 to 1. It is not absolutely continuous.

Misunderstanding: "Any bounded function with only finitely many discontinuities is Riemann integrable."

A function is Riemann integrable on [a,b] if and only if it is bounded and its set of discontinuities has Lebesgue measure zero (not merely finite or countable). The characteristic function of a fat Cantor set (which has positive measure but empty interior) is bounded and everywhere defined but not Riemann integrable.

Misunderstanding: "The Lebesgue integral is only useful when the Riemann integral fails."

The Lebesgue integral always agrees with the Riemann integral when the latter exists. Its utility even for Riemann-integrable functions is in the convergence theorems (MCT, DCT) and in abstract formulations (Fubini's theorem, Lᵖ spaces, Fourier analysis) that do not require the Euclidean setting at all.

Misunderstanding: "All subsets of ℝᵈ are Lebesgue measurable."

Vitali's construction (using the axiom of choice) produces a non-measurable subset of [0,1]. Such sets cannot be assigned a consistent translation-invariant measure. In practice, every set encountered without explicit appeal to the axiom of choice is measurable.

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Central paradox / key insight

The central paradox is this: the natural desire is for a measure that is (i) defined on all subsets of ℝᵈ, (ii) translation-invariant, and (iii) countably additive. The Banach–Tarski paradox and Vitali's non-measurable set construction show these three properties are simultaneously unachievable. The resolution — restricting to a σ-algebra of measurable sets — feels like a compromise, yet in practice it loses nothing: every set that ever arises without an explicit invocation of the axiom of choice is measurable.

The deeper insight is that the "right" formalization of size is not just a number attached to each set, but the triple (X, B, μ) — the sigma-algebra is part of the data of the theory, not an afterthought. And the key design parameter is countable (not finite, not uncountable) additivity. Tao makes this explicit in the comparison between Jordan outer measure (merely finitely subadditive, so ℚ has infinite Jordan outer measure) and Lebesgue outer measure (countably subadditive, so every countable set has Lebesgue outer measure zero). This one change — from finite to countable — is what makes the theory work under limits.

"The standard solution to the problem of measure has been to abandon the goal of measuring every subset E of ℝᵈ, and instead to settle for only measuring a certain subclass of 'non-pathological' subsets of ℝᵈ, which are then referred to as the measurable sets." — Tao, §1.1

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Important concepts

Measure space

A triple (X, B, μ) where X is a set, B is a σ-algebra (collection of subsets closed under complements and countable unions), and μ : B → [0,+∞] is a countably additive function with μ(∅) = 0.

σ-algebra (sigma-algebra)

A collection B of subsets of a set X satisfying: ∅ ∈ B; if E ∈ B then X\E ∈ B; and if E₁, E₂, … ∈ B then ⋃ₙEₙ ∈ B. The σ-algebra generated by a collection A is the smallest σ-algebra containing A.

Lebesgue outer measure

m∗(E) = inf{ ∑|Bₙ| : ⋃Bₙ ⊃ E, each Bₙ a box }. Satisfies m∗(∅)=0, monotonicity, and countable subadditivity; is not in general countably additive.

Lebesgue measurable set

E ⊂ ℝᵈ is Lebesgue measurable if for every ε > 0 there exists an open U ⊃ E with m∗(U\E) ≤ ε. Equivalently, E satisfies the Carathéodory criterion m∗(A) = m∗(A∩E) + m∗(A\E) for all A.

Null set (set of measure zero)

A set E with m(E) = 0. Countable sets are null sets; the Cantor set is an uncountable null set. Properties holding outside a null set hold almost everywhere (a.e.).

Simple function

A measurable function taking finitely many values: f = ∑cᵢ1_{Eᵢ}. The integral ∫f dμ = ∑cᵢμ(Eᵢ). Simple functions form the building blocks for the general Lebesgue integral.

Unsigned Lebesgue integral

For f : ℝᵈ → [0,+∞] measurable, ∫f dx = sup{ ∫g dx : 0 ≤ g ≤ f, g simple }. Always exists in [0,+∞].

Absolutely integrable function

f : X → ℂ is absolutely integrable (f ∈ L¹(μ)) if ∫|f| dμ < ∞. The Lebesgue integral of f is then defined via positive/negative (or real/imaginary) part decomposition.

Monotone convergence theorem (MCT)

If 0 ≤ f₁ ≤ f₂ ≤ … are measurable and fₙ → f pointwise, then ∫fₙ dμ → ∫f dμ. The fundamental tool for passing limits through integrals in the unsigned setting.

Dominated convergence theorem (DCT)

If fₙ → f pointwise a.e. and |fₙ| ≤ G a.e. with G absolutely integrable, then ∫fₙ dμ → ∫f dμ. The main tool for passing limits through integrals for signed/complex functions.

Fatou's lemma

For unsigned measurable fₙ ≥ 0: ∫lim inf fₙ dμ ≤ lim inf ∫fₙ dμ. Mass can be destroyed by a limit but not created.

Modes of convergence

Seven ways in which fₙ → f: (1) uniform; (2) L∞ (essential uniform); (3) almost uniform; (4) pointwise; (5) pointwise a.e.; (6) L¹; (7) in measure. These form a partial order, not a chain.

Uniform integrability

A sequence {fₙ} is uniformly integrable if supₙ‖fₙ‖{L¹} < ∞, ∫{|fₙ|≥M}|fₙ|→0 as M→∞, and ∫_{|fₙ|≤δ}|fₙ|→0 as δ→0. Uniform integrability is the condition under which convergence in measure ⟺ convergence in L¹.

Hardy–Littlewood maximal inequality

For f ∈ L¹(ℝ) and λ > 0, m({x : sup{h>0}(1/h)∫{[x,x+h]}|f| ≥ λ}) ≤ (1/λ)‖f‖_{L¹}. A weak-type (1,1) inequality; the foundational estimate of harmonic analysis.

Bounded variation (BV)

F : ℝ → ℝ has bounded variation if ‖F‖{TV} = sup{x₀<…<xₙ} ∑|F(xᵢ₊₁)−F(xᵢ)| < ∞. BV functions are differentiable a.e. BV = differences of monotone functions.

Absolute continuity

F is absolutely continuous on [a,b] if for every ε > 0 there exists δ > 0 such that ∑|F(bₖ)−F(aₖ)| ≤ ε whenever ∑(bₖ−aₖ) ≤ δ. Absolute continuity ⟺ F(b)−F(a) = ∫ₐᵇ F′(t) dt (second FTC holds).

Pre-measure

A countably additive function μ₀ on a Boolean algebra B₀. The Hahn–Kolmogorov theorem extends any σ-finite pre-measure uniquely to a measure on ⟨B₀⟩.

Carathéodory measurability

Given an outer measure μ∗ on X, a set E is Carathéodory measurable if μ∗(A) = μ∗(A∩E) + μ∗(A\E) for all A ⊂ X. The Carathéodory-measurable sets form a σ-algebra on which μ∗ is countably additive.

Fubini–Tonelli theorem

For σ-finite measure spaces and (unsigned or absolutely integrable) f : X×Y → ℂ, the double integral equals either iterated integral in any order. Requires σ-finiteness; without it, the theorem can fail.

Kolmogorov extension theorem

Given compatible inner-regular finite-dimensional probability measures μB on ∏{α∈B} Xα for every finite B ⊂ A, there exists a unique probability measure on the full product ∏{α∈A} Xα extending all μB. The foundation of the theory of stochastic processes.

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References and Web Links

Primary book and edition information

• Tao, T. An Introduction to Measure Theory. Graduate Studies in Mathematics, vol. 126. American Mathematical Society, 2011. ISBN 978-0-8218-6919-2.
  • AMS book page
  • Author's blog page for the book (with links to the PDF and errata)

Background and overview

• Tao's blog course notes (245A Real Analysis), from which the book derives
• Open Library record for the book
• Google Books entry

Secondary text used in the course

• Stein, E. and Shakarchi, R. Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton Lectures in Analysis, III. Princeton University Press, 2005. (Tao cites this as [StSk2005] throughout the book; his approach of treating Euclidean measure first before the abstract theory is directly inspired by Stein–Shakarchi.)

Companion volume

• Tao, T. An Epsilon of Room, Vol. I (Graduate Studies in Mathematics, vol. 117, AMS, 2010). Cited throughout as the sequel covering Hilbert and Banach spaces, Lᵖ theory, Fourier analysis, and distributions.
  • AMS page for An Epsilon of Room

Key results cited in the book

• Solovay, R. "A model of set theory in which every set of reals is Lebesgue measurable." Annals of Mathematics 92 (1970): 1–56. (Used in §1.1 regarding the necessity of the axiom of choice for non-measurable sets.)
• Melas, A. "The best constant for the centered Hardy–Littlewood maximal inequality." Annals of Mathematics 157 (2003): 647–688. (Referenced in §1.6 for the best constant in the maximal inequality.)

Reviews

• MAA Review by Mihaela Poplicher