Measure Theory and Modern Analysis

Measure Theory and Modern Analysis A Structured Introduction with Applications Rui Gonçalves Department of Civil and Georesources Engineering University of Porto 2025 ii ©2025 Rui Gonçalves Published by the author Porto, Portugal This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). ISBN 978-989-33-9046-7 Author affiliation: University of Porto iv Abstract This monograph presents a rigorous yet accessible introduction to measure theory and modern analysis. Beginning with outer measures and Carathéodory’s extension theorem, the text develops the foundational machinery leading to Lebesgue measure, measurable functions, and the Lebesgue integral. Advanced convergence theorems, Lp spaces, product measures, and change-of-variables techniques are explored in depth. The final chapters provide a concise introduction to Fourier analysis and the modern theory of concentration of measure. Throughout, historical notes illuminate the development of the subject, and exercises at the end of each chapter reinforce key ideas. v vi Contents Contents vii 1 Measure Spaces 1 1.1 Introduction ..................................... 1 1.2 Algebras and σ-Algebras .............................. 1 1.3 Measures ...................................... 2 1.4 Outer Measures and Carathéodory’s Criterion (Preview) ............ 2 1.5 A Fundamental Lemma .............................. 2 2 Outer Measures and Carathéodory’s Extension Theorem 5 2.1 Outer Measures ................................... 5 2.2 Carathéodory-Measurable Sets .......................... 5 2.3 Extension of Pre-Measures ............................. 6 3 Lebesgue Measure 9 3.1 Construction via Outer Measure ......................... 9 3.2 Open Sets and Countable Unions of Intervals .................. 9 3.3 Measurable Sets .................................. 10 3.4 Completion ..................................... 10 4 Measurable Functions 11 4.1 Characterizations .................................. 11 4.2 Simple Functions .................................. 12 4.3 Limits Superior and Inferior ............................ 12 5 The Lebesgue Integral 13 5.1 Integral of Simple Functions ............................ 13 5.2 Integral of Non-Negative Measurable Functions ................. 13 5.3 The Monotone Convergence Theorem ....................... 13 5.4 Fatou’s Lemma ................................... 14 5.5 The Dominated Convergence Theorem ...................... 14 5.6 The General Integral ................................ 15 vii viii CONTENTS 6 Convergence Theorems II 17 6.1 Convergence in Measure .............................. 17 6.2 Relations with Almost Everywhere Convergence ................. 17 6.3 Uniform Integrability ................................ 18 6.4 Vitali Convergence Theorem — Short Version .................. 18 6.5 Vitali Convergence Theorem — Full Version ................... 18 7 Real and Complex Integration 21 7.1 Complex-Valued Functions ............................ 21 7.2 Properties of the Integral ............................. 21 8LpSpaces 23 8.1 Definition and Basic Properties .......................... 23 8.2 Hölder’s Inequality ................................. 23 8.3 Minkowski’s Inequality ............................... 24 8.4 Completeness of Lp(Riesz–Fischer Theorem) .................. 24 9 Product Measures and the Fubini–Tonelli Theorems 27 9.1 Product σ-Algebras ................................. 27 9.2 Product Measures ................................. 27 9.3 Tonelli’s Theorem .................................. 27 9.4 Fubini’s Theorem .................................. 28 9.5 Counterexamples .................................. 28 10 Induced Measures and the Change of Variables Formula 29 10.1 Pushforward Measures and Distributions ..................... 29 10.2 Change of Variables: One Dimension ....................... 30 10.3 Jacobian and the Multivariable Formula ..................... 30 11 Probability and Random Variables 31 11.1 Probability as Measure ............................... 31 11.2 Distribution and Density .............................. 31 11.3 Expectation, Moments, Variance ......................... 31 11.4 Independence .................................... 31 11.5 Change of Variables in Probability ........................ 32 11.6 Conditional Expectation (Short Overview) .................... 32 12 Introduction to Fourier Analysis 33 12.1 The Fourier Transform on L1........................... 33 12.2 Basic Properties .................................. 33 12.3 Inversion and Plancherel’s Theorem ....................... 33 12.4 Convolution ..................................... 34 CONTENTS ix 13 Concentration of Measure 35 13.1 Hoeffding’s Inequality ............................... 35 13.2 McDiarmid’s Inequality .............................. 35 13.3 Talagrand-Type Inequality ............................. 35 13.4 Lévy’s Lemma on the Sphere ........................... 36 13.5 Gaussian Concentration .............................. 36 13.6 Poincaré and Log-Sobolev Inequalities ...................... 36 13.7 Applications ..................................... 36 14 Law of Large Numbers and the Central Limit Theorem 39 14.1 Introduction ..................................... 39 14.2 Weak Law of Large Numbers ........................... 39 14.3 Strong Law of Large Numbers ........................... 40 14.4 Lindeberg–Lévy Central Limit Theorem ..................... 40 14.5 Lindeberg and Lyapunov Generalizations .................... 41 14.6 Berry–Esseen Bound ................................ 41 14.7 Applications ..................................... 41 6 CHAPTER 2. OUTER MEASURES AND CARATHÉODORY’S EXTENSION THEOREM Denote by Mthe collection of such sets. Theorem 2.4. Misaσ-algebra, and the restriction µ=µ∗|Mis a complete measure. Proof (Expanded). We prove closure under complements and countable unions. Complements. Let A∈ M and E⊆X. Then µ∗(E) = µ∗(E∩A)+µ∗(E\A). By symmetry, µ∗(E) = µ∗(E∩Ac)+µ∗(E\Ac), so Ac∈ M. Countable unions. Let A1, A2,··· ∈ M and set A = SnAn . Fix E⊆X . We shall show µ∗(E) = µ∗(E∩A)+µ∗(E\A).(2.1) For each k , define Bk = Ak\Sj<k Aj . Then the Bk form a partition of A and are in M . For finite unions, repeated application of measurability yields µ∗(E) = µ∗(E∩A(N))+µ∗(E\A(N)), where A(N) = SN k=1 Ak . Letting N→ ∞ and using continuity from below of µ∗ on increasing sequences, we obtain (2.1). Completeness. Suppose N⊆A∈ M with µ∗(A)=0. Then for any E, µ∗(E) = µ∗(E\N) + µ∗(E∩N), but the last term must be zero. Hence Nis measurable. Remark 2.5 (Historical Note).Carathéodory’s key insight was that measurability can be defined by (2.1) rather than derived. This unifies all measure constructions: Lebesgue measure, Hausdorff measure, probability measures, and many others are all restrictions of suitable outer measures. 2.3 Extension of Pre-Measures Let Abe an algebra (not necessarily a σ-algebra) and µ0:A → [0,∞]be finitely additive. Definition 2.6. µ0is a pre-measure if for any disjoint An∈ A with SAn∈ A, µ0 [ n An!=X n µ0(An). 2.3. EXTENSION OF PRE-MEASURES 7 Define the outer measure induced by µ0: µ∗(E) = inf (∞ X k=1 µ0(Ak) : E⊆[ k Ak, Ak∈ A). Theorem 2.7 (Carathéodory Extension Theorem).If µ0 is a pre-measure on an algebra A , then the restriction of µ∗ to the Carathéodory-measurable sets is a measure extending µ0 . If µ0is σ-finite, the extension is unique. Proof (Expanded). Existence is immediate from the previous theorem: µ∗ is an outer measure, hence induces a complete measure µ. To show µ0 = µ on A , note that any A∈ A is covered by itself, so µ∗ ( A ) ≤µ0 ( A ). Conversely, any covering of A by sets in A must have total µ0 -measure at least µ0 ( A )by finite additivity. Thus µ∗(A)≥µ0(A). For uniqueness, assume µ and ν are two extensions on σ ( A ). Under σ -finiteness, standard arguments identify a determining class, showing µ=ν. Remark 2.8 (Historical Note).The uniqueness portion is where σ -finiteness is essential. Without it, pathological counterexamples exist (first observed by Vitali in different contexts). Carathéodory’s theorem remains the universal engine behind all measure constructions used in analysis today. Notes and Further Reading Carathéodory’s 1914 paper is one of the cornerstones of modern analysis. Its influence extends far beyond measure theory: the Carathéodory construction appears in geometric measure theory, probability, ergodic theory, and fractal geometry. Accessible expositions include Halmos (1950), Royden–Fitzpatrick, and Folland’s Chapter 1. Additional Exercises 1. Show that µ∗is monotone and countably subadditive. 2. Prove that the collection of Carathéodory-measurable sets is an algebra. 3. Prove uniqueness of the extension under σ-finiteness. 4. Let µ0 be Lebesgue pre-measure on finite unions of intervals. Show that the induced µ∗is the classical Lebesgue measure. 8 CHAPTER 2. OUTER MEASURES AND CARATHÉODORY’S EXTENSION THEOREM Chapter 3 Lebesgue Measure 3.1 Construction via Outer Measure Lebesgue’s idea (1902) was to define for any E⊆R: m∗(E) = inf (X k|Ik|:E⊆[ k Ik), where the infimum is taken over coverings by open intervals. This is an outer measure, and the measurable sets (in Carathéodory’s sense) are precisely the Lebesgue-measurable sets. Theorem 3.1 (Expanded).The function m∗defined above is an outer measure on R. Proof (Expanded). Monotonicity and m∗(∅) = 0 are immediate. For subadditivity, let E⊆SnEn . Cover each En by intervals Ink such that Pk|Ink| ≤ m∗(En)+ϵ2−n. Then the union of all Ink covers E, and m∗(E)≤X n,k |Ink| ≤ X n (m∗(En)+ϵ2−n). Letting ϵ→0yields the result. Remark 3.2 (Historical Note).Lebesgue arrived at m∗ through geometric intuition: length is approximated by covering with simple pieces. The abstraction to general outer measures came later. 3.2 Open Sets and Countable Unions of Intervals Theorem 3.3 (Expanded).Every open set U⊆R can be written as a countable, pairwise disjoint union of open intervals. Proof. For each x∈U, define a(x) = inf{y<x: [y, x]⊆U}, b(x) = sup{y > x : [x, y]⊆U}, 9 10 CHAPTER 3. LEBESGUE MEASURE and set Ix = ( a ( x ) , b ( x )). These intervals are either identical or disjoint. Since Q is countable, the collection of distinct intervals is countable: U=[ q∈Q∩U Iq. Remark 3.4 (Historical Note).This lemma plays a crucial role in the classical theory of functions of a real variable. It is essentially due to Borel. 3.3 Measurable Sets Definition 3.5. A set A⊆Ris Lebesgue measurable if m∗(E) = m∗(E∩A)+m∗(E\A)∀E⊆R. Theorem 3.6. The class of Lebesgue measurable sets is a σ -algebra containing all Borel sets. 3.4 Completion Theorem 3.7 (Expanded).Lebesgue measure m is complete: if N⊆A and m ( A ) = 0, then Nis measurable with m(N) = 0. Proof. Immediate from the Carathéodory construction: null sets automatically satisfy the splitting property. Notes and Further Reading Lebesgue’s original construction has been refined over the years for pedagogical clarity, but its essence remains unchanged. Modern treatments emphasize either the geometric viewpoint (as in the classical approach) or the abstract outer-measure formulation. Bogachev’s two-volume monograph provides the most exhaustive modern account. Additional Exercises 1. Verify that m∗defines an outer measure. 2. Prove that every closed set is measurable. 3. Prove that every set of measure zero is measurable. 4. Show that the half-open interval [a, b)has measure b−a. Chapter 4 Measurable Functions 4.1 Characterizations The notion of measurability generalizes continuity: measurable functions are those compatible with the underlying σ-algebra. Definition 4.1. Let ( X, M )be a measurable space and ( Y, N )another. A function f : X→ Yis measurable if f−1(B)∈ M ∀B∈ N. In the case Y = R with its Borel σ -algebra B , we have the following classical characterization. Theorem 4.2 (Equivalent Conditions).Let f:X→R. The following are equivalent: 1. fis measurable, 2. f−1((−∞, a)) ∈ M for all a, 3. f−1((a, ∞)) ∈ M for all a, 4. f−1([a, ∞)) ∈ M for all a. Proof (Expanded). The equivalences follow from the fact that such families of sets generate the Borel σ-algebra. For instance, (−∞, a) = [ q<a, q∈Q (−∞, q), so measurability with respect to rational cuts suffices. Remark 4.3 (Historical Note).Lebesgue recognized early that measurability only depends on preimages of intervals; the abstraction through generating classes came later, notably through the work of Dynkin and Carathéodory in the context of monotone classes and λ-systems. 11 12 CHAPTER 4. MEASURABLE FUNCTIONS 4.2 Simple Functions Definition 4.4. A function s:X→Ris simple if it takes finitely many values: s= n X k=1 ak1Ak, Ak∈ M. Lemma 4.5 (Approximation).If f : X→ [0 ,∞ ]is measurable, then there exists an increasing sequence of simple functions snsuch that sn↑fpointwise. Proof (Expanded). Define sn(x) = k 2nif k 2n≤f(x)<k+ 1 2n. Then each snis simple, sn≤f, and sn↑f. Remark 4.6 (Historical Note).This idea goes back to Lebesgue’s construction of the integral: the use of dyadic partitions (powers of 2) is purely technical but highlights the role of rational approximations in the theory. 4.3 Limits Superior and Inferior For measurable functions fn , the pointwise lim sup fn and lim inf fn are measurable because supremum and infimum of measurable functions remain measurable (proved via level sets). Notes and Further Reading Measurable functions occupy the central position in analysis. Halmos (1950) and Folland (1999) offer elegant treatments. Dynkin’s π - λ theorem underpins many arguments related to measurability. Additional Exercises 1. Show that if fis measurable and gis continuous, then g◦fis measurable. 2. Prove that supnfnand infnfnare measurable. 3. Show that if fn↑f, then lim supfn=f. 4. Construct a sequence of measurable functions whose limit is not bounded on any set of positive measure. Chapter 5 The Lebesgue Integral 5.1 Integral of Simple Functions Let s=Pn k=1 ak1Akwith ak≥0. Define Zs dµ = n X k=1 akµ(Ak). 5.2 Integral of Non-Negative Measurable Functions For f≥0, define Zf dµ = supZs dµ :0≤s≤f, s simple. Lemma 5.1. If fn↑f, then Rfndµ ↑Rf dµ. Proof. Straight from the definition, but see the Monotone Convergence Theorem below for a deeper justification. 5.3 The Monotone Convergence Theorem Theorem 5.2 (Beppo Levi).If fn↑f, all measurable and non-negative, then Zfndµ ↑Zf dµ. Proof (Expanded). Let sn be simple with sn≤fn increasing to fn . Then sn↑f ; hence by definition of the integral, Zf dµ = supZsndµ. Since sn≤fn, we have Rsndµ ≤Rfndµ. Thus Rf dµ ≤lim inf Rfndµ. The reverse inequality follows by monotonicity: fn≤f⇒Rfndµ ≤Rf dµ. 13 14 CHAPTER 5. THE LEBESGUE INTEGRAL Remark 5.3 (Historical Note).This theorem first appeared in Beppo Levi’s 1906 work. It lies at the core of modern integration and is a powerful tool in analysis, PDE, probability, and functional analysis. 5.4 Fatou’s Lemma Theorem 5.4. For measurable fn≥0, Zlim inf n→∞ fndµ ≤lim inf n→∞ Zfndµ. Proof (Expanded). Let gk= infn≥kfn, so gk↑lim inf fn. Fatou’s lemma is equivalent to Zgkdµ ↑Zlim inf fndµ. Since gk≤fnfor all n≥k,Zgkdµ ≤Zfndµ, so Zgkdµ ≤inf n≥kZfndµ. Taking k→ ∞ gives the result. Remark 5.5 (Historical Note).Fatou (1906) introduced this inequality in the study of trigonometric series and harmonic analysis. It became foundational in the modern theory. 5.5 The Dominated Convergence Theorem Theorem 5.6 (Lebesgue Dominated Convergence).If fn→f almost everywhere, |fn|≤g with gintegrable, then Zfndµ →Zf dµ. Proof (Fully Expanded). Let hn = |fn−f| . Since fn→f almost everywhere, hn→ 0almost everywhere. Note that hn≤2g. Set uk = supn≥khn . Then uk↓ 0almost everywhere and uk≤ 2 g . By the Monotone Convergence Theorem applied to 2g−uk, we obtain Zukdµ ↓0. Now fix ε > 0. Choose ksuch that Rukdµ < ε. For all n≥k,hn≤uk, hence Zfndµ −Zf dµ=Z(fn−f)dµ≤Z|fn−f|dµ ≤Zukdµ < ε. 5.6. THE GENERAL INTEGRAL 15 Thus Rfndµ →Rf dµ. Remark 5.7 (Historical Note).Lebesgue’s 1910 paper contains a version of this result, though not in full generality. The conceptual role of domination was clarified later in the development of real analysis. 5.6 The General Integral For an integrable real function f , write f = f+−f− , where f± = max ( ±f, 0). We say f is integrable if both f+and f−have finite integrals. Notes and Further Reading The dominance of Lebesgue’s theory over Riemann’s stems largely from the convergence theorems in this chapter. Beppo Levi (1906), Fatou (1906), and Lebesgue (1910) laid the foundation of modern integration theory. Additional Exercises 1. Prove that integration is monotone: f≤g⇒Rf≤Rg. 2. Construct a sequence where Fatou’s lemma is strict. 3. Find an example where dominated convergence fails due to lack of a dominating function. 4. Show that fn→fin L1implies a subsequence converges a.e. 22 CHAPTER 7. REAL AND COMPLEX INTEGRATION Proof (Expanded). Write f=u+iv. By the triangle inequality in C, Zf≤Zu+Zv≤Z|u|+Z|v| ≤ Z|f|. Linearity If fand gare integrable and α, β ∈C, then Z(αf +βg)dµ =αZf dµ +βZg dµ. Stability Under L1Convergence If fn→fin L1, then Zfndµ →Zf dµ. This follows immediately from the dominated convergence theorem applied to |fn−f|. Notes and Further Reading Complex integration in the Lebesgue sense is an immediate extension of the real theory. Standard references include Folland, Royden–Fitzpatrick, and the classical works of Lebesgue and Riesz. Additional Exercises 1. Show that if f∈L1, then ℜfand ℑfalso lie in L1. 2. Prove the inequality |Rf| ≤ R|f|directly for real functions. 3. Give an example of a complex-valued function that is integrable but not absolutely integrable. 4. Verify that L1convergence implies convergence of integrals. Chapter 8 LpSpaces 8.1 Definition and Basic Properties For 1≤p<∞, define ∥f∥p=Z|f|pdµ1/p , and for p=∞, ∥f∥∞= inf{M:|f|≤Ma.e.}. Definition 8.1. Lp ( µ )is the space of (equivalence classes of) measurable functions with finite p-norm. Remark 8.2 (Historical Note).Lebesgue introduced Lp spaces in his foundational work; their structural importance was clarified by Riesz, Fischer, and Banach. The study of Lp spaces is central to functional analysis, PDE, harmonic analysis, and probability. 8.2 Hölder’s Inequality Theorem 8.3 (Hölder).If 1< p, q < ∞with 1 p+1 q= 1, then Z|fg|≤∥f∥p∥g∥q. Proof (Expanded via Young’s Inequality). Young’s inequality states that for a, b ≥0, ab ≤ap p+bq q. Let u=|f|/∥f∥pand v=|g|/∥g∥q. Then |fg|=∥f∥p∥g∥quv ≤ ∥f∥p∥g∥qup p+vq q. 23 24 CHAPTER 8. LpSPACES Integrate both sides: Z|fg|≤∥f∥p∥g∥q1 pZup+1 qZvq=∥f∥p∥g∥q. Remark 8.4 (Historical Note).Hölder’s inequality (1888) predates Lebesgue’s integration theory but became foundational after 1902. Hölder himself used it for series and normed spaces long before the formalization of Lptheory. 8.3 Minkowski’s Inequality Theorem 8.5 (Minkowski).For 1≤p < ∞, ∥f+g∥p≤ ∥f∥p+∥g∥p. Proof (Expanded). For p>1, apply Hölder with the pair (p, q): |f+g|p=|f+g||f+g|p−1≤(|f|+|g|)|f+g|p−1. Thus, Z|f+g|p≤Z|f||f+g|p−1+Z|g||f+g|p−1≤ ∥f∥p∥f+g∥p−1 p+∥g∥p∥f+g∥p−1 p. Divide by ∥f+g∥p−1 pto finish. Remark 8.6 (Historical Note).Minkowski (1896) introduced this inequality in the context of geometry of numbers. It predates Lebesgue but became essential in establishing that Lp is a normed space. 8.4 Completeness of Lp(Riesz–Fischer Theorem) Theorem 8.7 (Riesz–Fischer). Lp ( µ )is complete: every Cauchy sequence converges in Lp to a limit in Lp. Proof (Expanded). Let {fn}be Cauchy in Lp. Then ∥fn−fm∥p→0as n, m → ∞. Step 1: Extract an almost everywhere Cauchy subsequence. Choose nk so that ∥fnk−fnk+1 ∥p<2−k. Set g=fn1+∞ X k=1 (fnk+1 −fnk). This series converges a.e., giving a candidate limit. 8.4. COMPLETENESS OF Lp(RIESZ–FISCHER THEOREM) 25 Step 2: Show g∈Lp.Using Minkowski and the geometric decay, ∥g∥p≤ ∥fn1∥p+∞ X k=1 ∥fnk+1 −fnk∥p<∞. Step 3: Show fnk→gin Lp.Again by the triangle inequality and geometric decay: ∥fnk−g∥p→0. Step 4: Extend convergence to the full sequence. Cauchyness implies any subsequence’s limit is the full limit. Remark 8.8 (Historical Note).The Riesz–Fischer theorem (1907) was a cornerstone in establishing L2 as a Hilbert space, later extended to Lp spaces. It marked the birth of functional analysis as a discipline. Notes and Further Reading Lp spaces lie at the core of modern analysis. Their geometry is rich: L2 is Hilbertian, while Lp for p = 2 is strictly convex but not inner-product based. The foundational works of Riesz, Fischer, and Banach form the backbone of their theory. Additional Exercises 1. Prove that ∥f∥p→ ∥f∥qas p→qfor fixed fon finite measure spaces. 2. Give an example of a sequence bounded in L1but not uniformly integrable. 3. Show that Lpis separable for 1≤p < ∞. 4. Prove that ∥fn−f∥p→0implies a subsequence converges a.e. 26 CHAPTER 8. LpSPACES Chapter 9 Product Measures and the Fubini–Tonelli Theorems 9.1 Product σ-Algebras Given measurable spaces ( X, M )and ( Y, N ), the product σ -algebra M⊗N is the smallest σ-algebra containing measurable rectangles A×B. 9.2 Product Measures Let µ and ν be σ -finite measures on X and Y . There exists a unique measure µ×ν on M⊗Nsatisfying (µ×ν)(A×B) = µ(A)ν(B). Remark 9.1 (Historical Note).The development of product measures was crucial for Kolmogorov’s construction of probability spaces of stochastic processes. Fubini’s theorem predates Lebesgue; Tonelli extended it to non-negative functions. 9.3 Tonelli’s Theorem Theorem 9.2 (Tonelli).If f:X×Y→[0,∞]is measurable, then ZXZY f(x, y)dν(y)dµ(x) = ZYZX f(x, y)dµ(x)dν(y) = ZX×Y f d(µ×ν). Proof (Expanded). First verify the identity for simple functions Pak 1 Ak×Bk . Then extend to monotone limits of simple functions using the Monotone Convergence Theorem. Remark 9.3 (Historical Note).Tonelli (1909) realized that non-negative functions are always integrable in the “extended” sense of the integral, making the iterated integrals identical without further assumptions. 27 28 CHAPTER 9. PRODUCT MEASURES AND THE FUBINI–TONELLI THEOREMS 9.4 Fubini’s Theorem Theorem 9.4 (Fubini).If fis integrable on X×Y, then the iterated integrals exist and ZX×Y f=ZXZY f=ZYZX f. Proof (Expanded). Since f∈L1 ( µ×ν ), both f+ and f− are integrable. Apply Tonelli to f+ and f−separately, then subtract. Remark 9.5 (Historical Note).Fubini (1907) studied conditions for switching the order of integration. Lebesgue later incorporated it into his general theory. 9.5 Counterexamples If f is not integrable, the iterated integrals may differ or may be infinite in different orders. Classical examples appear in every analysis text, highlighting the necessity of integrability for Fubini. Notes and Further Reading Fubini–Tonelli theory is foundational in PDE, Fourier analysis, and probability. A modern perspective appears in Rudin’s Real and Complex Analysis and Folland. Additional Exercises 1. Give an example where Fubini fails because f /∈L1. 2. Prove Tonelli’s theorem for simple functions in full detail. 3. Show that the product of σ-finite measures is unique. 4. Verify Fubini’s theorem for f∈L1(X×Y). Chapter 10 Induced Measures and the Change of Variables Formula 10.1 Pushforward Measures and Distributions Let ( X, M, µ )be a measure space and f : X→Y a measurable function into another measurable space (Y, N). Definition 10.1. The pushforward measure (or induced measure) f#µis defined by (f#µ)(B)=µ(f−1(B)), B ∈ N. Proposition 10.2. f#µis a measure on (Y, N). Proof. Since f−1 ( ∅ ) = ∅ and preimages preserve disjoint unions, countable additivity follows immediately: (f#µ) [ k Bk!=µ [ k f−1(Bk)!=X k µ(f−1(Bk)). Remark 10.3 (Historical Note).Pushforward measures arise naturally in probability theory: the law of a random variable is the pushforward of the underlying probability measure. The concept appears implicitly in Lebesgue’s work and explicitly in Kolmogorov’s axiomatization. Distributions of Real-Valued Functions If f:X→Ris measurable, its distribution is the measure f#µon the Borel sets of R. The cumulative distribution function (CDF) is F(t) = µ(f≤t). If F is absolutely continuous, then F′ ( t ) = ρ ( t )almost everywhere, where ρ is the density of the induced measure. 29 30 CHAPTER 10. INDUCED MEASURES AND THE CHANGE OF VARIABLES FORMULA 10.2 Change of Variables: One Dimension Theorem 10.4 (Change of Variables, 1D).Let f : R→R be C1 and strictly monotone. Let gbe integrable. Then ZRg(f(x))|f′(x)|dx =ZRg(y)dy. Proof (Expanded). Since f is a C1 bijection onto its image with non-vanishing derivative, the substitution y = f ( x )gives dy = f′ ( x ) dx . A rigorous proof is obtained by approximating g with simple functions and using monotone convergence. 10.3 Jacobian and the Multivariable Formula Let f:Rn→RnbeaC1diffeomorphism. Theorem 10.5 (Change of Variables in Rn).If gis integrable, then ZRng(f(x))|detDf(x)|dx =ZRng(y)dy. Proof (Expanded). On each small cube Q of a fine partition of Rn , the mapping f is close to its differential, so m(f(Q)) ≈ |detDf(ξQ)|m(Q), where ξQ∈Q. Summing over all cubes and refining the partition gives the result. Notes and Further Reading Pushforward measures are fundamental in probability, dynamical systems, optimal transport, and geometric measure theory. The multidimensional change-of-variables formula is a classical result with roots in the work of Jacobi and its rigorous justification through Lebesgue’s theory. Additional Exercises 1. Compute the pushforward of Lebesgue measure under f(x)=x2. 2. Show that f#µis complete if µis complete. 3. Verify the substitution formula for affine maps f(x)=ax +b. 4. Compute the Jacobian determinant of the polar coordinate map. Chapter 11 Probability and Random Variables 11.1 Probability as Measure Let (Ω,F,P)be a probability space: Pis a measure with P(Ω) = 1. Definition 11.1. Arandom variable is a measurable function X: Ω →R. 11.2 Distribution and Density The distribution of Xis the pushforward measure X#P, and its CDF is FX(t)=P(X≤t). If FXis absolutely continuous, then Xhas a density ρand FX(t) = Zt −∞ ρ(s)ds. 11.3 Expectation, Moments, Variance E[X] = ZΩ X dP. If X∈L2, then Var(X)=E[X2]−(E[X])2. Higher moments are defined similarly. 11.4 Independence Random variables Xand Yare independent if P(X∈A, Y ∈B) = P(X∈A)P(Y∈B) 31 38 CHAPTER 13. CONCENTRATION OF MEASURE Chapter 14 Law of Large Numbers and the Central Limit Theorem 14.1 Introduction Random phenomena often exhibit remarkable regularity when observed in aggregate. Two classical results formalize this idea: the Law of Large Numbers (LLN), describing the convergence of sample averages, and the Central Limit Theorem (CLT), describing their asymptotic distribution. These theorems underpin modern statistics, probability, and data science, linking randomness with deterministic behavior in the limit. Historically, Bernoulli proved the first version of the LLN (1713), while Laplace initiated the CLT. The fully rigorous forms appeared only with the development of measure theory and modern integration. 14.2 Weak Law of Large Numbers Let X1, X2, . . . be independent identically distributed (i.i.d.) random variables with mean µ=E[X1](assuming it exists). Define the sample average Xn=1 n n X k=1 Xk. Theorem 14.1 (Weak Law of Large Numbers).If Var(X1)<∞, then Xn P −→ µ. Proof (via Chebyshev). Since the Xk are independent and identically distributed, Var ( Xn ) = σ2/n, where σ2= Var(X1). Chebyshev’s inequality gives P|Xn−µ|> ε≤σ2 nε2→0. 39 40 CHAPTER 14. LAW OF LARGE NUMBERS AND THE CENTRAL LIMIT THEOREM Hence Xn→µin probability. Remark 14.2 (Historical Note).Bernoulli’s original (1713) proof considered Bernoulli trials and was non-quantitative. Chebyshev (1867) produced the first general inequality, making the modern LLN possible. 14.3 Strong Law of Large Numbers The strong form asserts almost sure convergence. Theorem 14.3 (Strong Law of Large Numbers).Let X1, X2, . . . be i.i.d. with E [ |X1| ] <∞ . Then Xn a.s. −−→ µ. Sketch. For square-integrable variables, Kolmogorov’s inequality and the Borel–Cantelli lemma imply X n P|Sn|> nε<∞, where Sn = X1 + ··· + Xn . Hence |Sn|/ ( nε ) → 0almost surely for all ε > 0. General integrable variables follow by truncation and dominated convergence. Remark 14.4 (Historical Note).The strong law in full generality is due to Kolmogorov (1929), whose framework unified probability theory on measure-theoretic foundations. 14.4 Lindeberg–Lévy Central Limit Theorem Theorem 14.5 (Classical Central Limit Theorem).Let X1, X2, . . . be i.i.d. with mean µ and variance σ2∈(0,∞). Then √n(Xn−µ) σD −→ N(0,1). Idea of Proof. Let φ(t)=E[eitX1]be the characteristic function. Then Eheit√n(Xn−µ)/σi=φt σ√ne−itµ/(σ√n)n . A Taylor expansion of φat 0yields φ(h) = 1 + iµh −1 2σ2h2+o(h2), from which the limit characteristic function is e−t2/2, the Fourier transform of N(0,1). Remark 14.6 (Historical Note).Lindeberg (1922) and Lévy (1925) refined earlier versions by Laplace and Lyapunov. Characteristic functions became the canonical tool via Lévy’s work. 14.5. LINDEBERG AND LYAPUNOV GENERALIZATIONS 41 14.5 Lindeberg and Lyapunov Generalizations The independence need not be identical, provided a regularity condition is satisfied. Let X1, X2, . . . be independent with means µkand variances σ2 k. Define s2 n= n X k=1 σ2 k. Definition 14.7 (Lindeberg Condition).For every ε>0, 1 s2 n n X k=1 E(Xk−µk)21{|Xk−µk|>εsn}−→ 0. Theorem 14.8 (Lindeberg–Feller CLT).If s2 n→ ∞ and the Lindeberg condition holds, then 1 sn n X k=1 (Xk−µk)D −→ N(0,1). Remark 14.9.Lyapunov’s condition, 1 s2+δ n n X k=1 E|Xk−µk|2+δ→0, for some δ > 0is stronger but easier to verify. 14.6 Berry–Esseen Bound The CLT provides an asymptotic distribution; the Berry–Esseen theorem quantifies the rate of convergence. Theorem 14.10 (Berry–Esseen).Let X1, . . . , Xn be i.i.d. with mean µ , variance σ2 , and third absolute moment ρ=E|X1−µ|3<∞. Then for all x, PSn−nµ σ√n≤x−Φ(x)≤Cρ σ3√n, where Φis the standard normal CDF and Cis a universal constant. Remark 14.11 (Historical Note).Berry (1941) and Esseen (1942) established versions of this inequality. Sharp constants remain an active area of research. 14.7 Applications Statistics Sample means, maximum likelihood estimators, and confidence intervals rely on LLN and CLT as foundational asymptotic tools. 42 CHAPTER 14. LAW OF LARGE NUMBERS AND THE CENTRAL LIMIT THEOREM Stochastic Processes Martingales, random walks, and Brownian motion are deeply connected to limit theorems; Donsker’s theorem generalizes the CLT to functional form. Numerical Simulation Monte Carlo methods depend on LLN for convergence and on CLT for error estimation. Additional Exercises 1. Prove the Weak LLN using Markov’s inequality for non-negative variables. 2. Show that if E|X1|<∞but Var(X1)=∞, the Weak LLN may fail. 3. Verify the characteristic function convergence in the classical CLT for Gaussian variables as a warm-up. 4. Prove Lyapunov’s CLT under the assumption δ= 1. 5. Give an example where Lindeberg’s condition holds but Lyapunov’s does not. 6. State and prove a version of the Strong LLN for bounded random variables.