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Analysis, Measure, and Probability: a Visual Introduction

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Analysis, Measure, and Probability: a Visual Introduction Analysis, Measure, and Probability: A visual introduction Marcus Pivato March 28, 2003 ii Contents Preface ......................................... x 1 Basic Ideas 1 1.1 Preliminaries ..................................... 1 1.1(a) The concept of size .............................. 1 1.1(b) Sigma Algebras ................................ 2 1.1(c) Measures ................................... 6 1.2 Basic Properties and Constructions ......................... 17 1.2(a) Continuity/Monotonicity Properties .................... 17 1.2(b) Sets of Measure Zero ............................. 18 1.2(c) Measure Subspaces .............................. 20 1.2(d) Disjoint Unions ................................ 20 1.2(e) Finite vs. sigma-finite ............................ 21 1.2(f) Sums and Limits of Measures ........................ 22 1.2(g) Product Measures .............................. 22 1.2(h) Diagonal Measures .............................. 23 1.3 Mappings between Measure Spaces ......................... 23 1.3(a) Measurable Functions ............................ 23 1.3(b) Measure-Preserving Functions ........................ 27 1.3(c) ‘Almost everywhere’ ............................. 29 1.3(d) A categorical approach ............................ 33 2 Construction of Measures 33 2.1 Outer Measures ................................... 33 2.1(a) Covering Outer Measures .......................... 39 2.1(b) Premeasures .................................. 43 2.1(c) Metric Outer Measures ........................... 48 2.2 More About Stieltjes Measures ........................... 51 2.3 Signed and Complex-valued Measures ....................... 57 2.4 The Space of Measures ................................ 62 2.4(a) Introduction ................................. 62 2.4(b) The Norm Topology ............................. 63 2.4(c) The Weak Topology ............................. 65 iii iv CONTENTS 2.5 Disintegrations of Measures ............................. 66 3 Integration Theory 69 3.1 Construction of the Lebesgue Integral ....................... 69 3.1(a) Simple Functions ............................... 71 3.1(b) Definition of Lebesgue Integral (First Approach) ............. 73 3.1(c) Basic Properties of the Integral ....................... 74 3.1(d) Definition of Lebesgue Integral (Second Approach) ............ 79 3.2 Limit Theorems ................................... 84 3.3 Integration over Product Spaces ........................... 86 4 Functional Analysis 94 4.1 Functions and Vectors ................................ 94 4.1(a) C, the spaces of continuous functions ................... 95 4.1(b) Cn, the spaces of differentiable functions .................. 96 4.1(c) L, the space of measurable functions .................... 97 4.1(d) L1, the space of integrable functions .................... 97 4.1(e) L∞, the space of essentially bounded functions .............. 98 4.2 Inner Products (infinite-dimensional geometry) .................. 99 4.3 L2 space ....................................... 101 4.4 Orthogonality ..................................... 102 4.4(a) Trigonometric Orthogonality ......................... 103 4.5 Convergence Concepts ................................ 106 4.5(a) Pointwise Convergence ............................ 106 4.5(b) Almost-Everywhere Convergence ...................... 109 4.5(c) Uniform Convergence ............................ 110 4.5(d) L∞ Convergence ............................... 113 4.5(e) Almost Uniform Convergence ........................ 115 4.5(f) L1 convergence ................................ 115 4.5(g) L2 convergence ................................ 118 4.5(h) Lp convergence ................................ 122 5 Information 125 5.1 Sigma Algebras as Information ........................... 125 5.1(a) Refinement and Filtration .......................... 130 5.1(b) Conditional Probability and Independence ................. 132 5.2 Conditional Expectation ............................... 136 5.2(a) Blurred Vision... ............................... 136 5.2(b) Some Examples ............................... 137 5.2(c) Existence in L2 ................................ 142 5.2(d) Existence in L1 ................................ 142 5.2(e) Properties of Conditional Expectation ................... 143 CONTENTS v 6 Measure Algebras 147 6.1 Sigma Ideals ...................................... 147 6.2 Measure Algebras ................................... 148 6.2(a) Algebraic Structure .............................. 148 6.2(b) Metric Structure ............................... 152 vi CONTENTS List of Figures 1.1 The notions of cardinality, length, area, volume, frequency, and probability are all examples of measures. ................................ 1 1.2 (A) A sigma algebra is closed under countable unions; (B) A sigma algebra is closed under countable intersections. ........................ 2 1.3 P is a partition of X. ................................ 4 1.4 The sigma-algebra generated by a partition: Partition the square into four smaller squares, so P = {P1, P2, P3, P4}. The corresponding sigma-algebra contains 16 elements. ................................ 4 1.5 Partition Q refines P if every element of P is a union of elements in Q. .... 4 1.6 The product sigma-algebra. ............................. 6 1.7 The Haar Measure: The Haar measure of U is the same as that of U + ~v .. 8 1.8 The Hausdorff measure. ............................... 9 1.9 Stieltjes measures: The measure of the set U is the amount of height “accu- mulated” by f as we move from one end of U to the other. ........... 11 1.10 (A) Outer regularity: The measure of B is well-approximated by a slightly larger open set U. (B) Inner regularity: The measure of B is well-approximated by a slightly smaller compact set K. ........................... 12 1.11 Density Functions: The measure of U can be thought of as the ‘area under the curve’ of the function f in the region over U. ................. 13 1.12 Probability Measures: X is the ‘set of all possible worlds’. Y ⊂ X is the set of all worlds where it is raining in Toronto. .................... 14 1.13 The Cantor set .................................... 18 1.14 The diagonal measure ................................ 23 1.15 Measurable mappings between partitions: Here, X and Y are partitioned into grids of small squares. The function f is measurable if each of the six squares covering Y gets pulled back to a union of squares in X. ............. 25 −1 2 1.16 Elements of pr1,2 (I ) look like“vertical fibres”. ................. 26 1.17 Measure-preserving mappings ............................ 27 1.18 (A) A special linear transformation on R2 maps a rectangle to parallelogram with the same area. (B) A special linear transformation on R3 maps a box to parallelopiped with the same volume. ....................... 28 1.19 The projection from I2 to I is measure-preserving. ................ 28 vii viii LIST OF FIGURES 2.1 The intervals (a1, b1], (a2, b2],..., (a8, b8] form a covering of U. ......... 34 2.2 (A) E1 ⊃ E2 ⊃ E3 ⊃ ... are neighbourhoods of the identity element e ∈ G. 14 (B) {gk · E3}k=1 is a covering of E1. ........................ 40 2.3 The product sigma-algebra is a prealgebra .....................R 44 2.4 Antiderivatives: If f(x) = arctan(x), then µ [a, b] = b 1 dx. ....... 52 f a 1+x2 2.5 The Heaviside step function. ............................ 52 2.6 The floor function f(x) = bxc. ........................... 53 2.7 The Devil’s staircase. ................................. 53 2.8 The fibre measure over a point. ........................... 67 2.9 The fibre of the function f over the point y. ................... 69 3.1 (A) f = f + − f −; (B) A simple function. ................... 70 3.2 (A) Repartitioning a simple function. (B) Repartitioning two simple functions to be compatible. .................................. 72 3.3 Approximating f with simple functions. ...................... 73 3.4 The Monotone Convergence Theorem ....................... 76 3.5 Lebesgue’s Dominated Convergence Theorem ................... 85 3.6 The fibre of a set. ................................... 87 3.7 The fibre of a set. ................................... 89 3.8 Z(n,m) = X(n) × Y(m). ................................. 91 4.1 We can think of a function as an “infinite-dimensional vector” .......... 94 4.2 (A) We add vectors componentwise: If u = (4, 2, 1, 0, 1, 3, 2) and v = (1, 4, 3, 1, 2, 3, 1), then the equation “w = v + w” means that w = (5, 6, 4, 1, 3, 6, 3). (B) We add two functions pointwise: If f(x) = x, and g(x) = x2 − 3x + 2, then the equation 2 “h = f + g” means that h(x) = f(x) +Rg(x) = x − 2x + 2 for every x. ..... 95 4.3 The L1 norm of f is defined: kfk = |f(x)| dµ[x]. .............. 98 1 X 4.4 The L∞ norm of f is defined: kfk = ess sup |f(x)|. ............ 98 qR ∞ x∈X 4.5 The L2 norm of f: kfk = |f(x)|2 dµ[x] .................. 101 2 X 4.6 Four Haar basis elements: H1, H2, H3, H4 ..................... 102 4.7 Seven Wavelet basis elements: W1,0; W2,0, W2,1; W3,0, W3,1, W3,2, W3,3 .. 103 4.8 C1, C2, C3, and C4; S1, S2, S3, and S4 ..................... 104 4.9 The lattice of convergence types. Here, “A=⇒B” means that convergence of type A implies convergence of type B, under the stipulated conditions. ..... 106 4.10 The sequence {f1, f2, f3,...} converges pointwise to the constant 0 function. Thus, if we pick some random points w, x, y, z ∈ X, then we see that lim fn(w) = n→∞ 0, lim fn(x) = 0, lim fn(y) = 0, and lim fn(z) = 0. .............. 107 n→∞ n→∞ n→∞ 4.11 If g (x) = 1 , then the sequence {g , g , g ,...} converges pointwise
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