Index

C₀ℝ, 177

C_cΩ, 138

F_δ, 19

G_δ, 19

L^p-Spaces, 127

V_{a, b}(φ), 146

σ-algebra generated by, 43

σ-filed, 60

σ-finite measure space, 39

σ-algebra, 38

k-cells, 58

x-section, 162, 169

y-section, 162, 169

L^pΩ, 129

Lp(Ω,MΩ,m), 129

L(X), 114

L(μ), 114

L∞(X,A,μ), 128

L∞(μ), 128

Lp(X), 127

Lp(X,A,μ), 127

Lp(Ω), 129

Lp(Ω,MΩ,m), 129

Lp(μ), 127

absolutely continuous, 103, 144

algebra, 38

almost everywhere, 67

convergence, 66

Borel σ-algebra, 43, 76

Borel measure, 46

Borel sets, 43

bounded

above, 12

below, 12

bounded intervals, 11

bounded variation, 146

canonical representation, 71

Cantor set, 22

Chandrasekharan, 186

characteristic function, 10, 58

closed set, 43

complete measure, 48

completion, 50

complex measurable function, 56

conditional expectation, 105

continuous, 56

convergence

in measure, 71

converges

almost everywhere, 66

pointwise, 65

convolution, 175, 192

countable additivity, 30

countable family, 11

counter examples, 172

counting measure, 40

Darboux-Riemann integral, 3

DCT, 121, 123

De-Morgan, 30

dense, 43

Dirac measure, 40

Dirichlet's function, 2, 7, 10

disjoint family, 23

distribution function, 61, 104

cumulative, 61

dominated convergence theorem, 121, 123

Egoroff's theorem, 69

elementary set, 164

essentially bounded, 127

even function, 180

events, 60

Fatou's lemma, 101, 155

finite measure, 39

finite measure space, 39

Fourier

coefficient, 158

inversion theorem, 187

transform, 177

Fourier-Plancherel

theorem, 196

transform, 183, 194

Fubini's theorem, 169, 171

fundamental theorem of Lebesgue integration, 142, 146, 151

greatest lower bound, 12

Hölder's inequality, 131

improper integral, 137

indefinite integral, 141

infimum, 13, 64

integrability, 113, 114

integrable, 114

integral, 113, 114

integral of φ

over E, 82

over X, 81

integral of f, 90

over E, 90

integration on product spaces, 161

inversion theorem, 187

Jensen's inequality, 129

least upper bound, 12

Lebesgue

integrable, 119

integral, 11, 115

measurable, 25

measurable sets, 16

measure, 30, 42

outer measure, 15, 42

Lebesgue integrable function, 114

length of the curve, 146

limit

inferior, 64

superior, 64

locally compact, 138

lower

bound, 12

Darboux integral, 3

Darboux sum, 3

MCT, 102

measurable, 59

Borel, 57

extended real valued, 57

function, 56

Lebesgue, 57

rectangles, 162

sets, 38

space, 38

measure, 38

induced by φ and μ, 88

induced by f and μ, 60, 100

positive, 39

space, 38

zero, 8

mesh, 5

Minkowski inequality, 132

modulus, 63

monotone

class, 164

convergence theorem, 97

monotonicity, 19, 39

negative part, 63

non-Lebesgue measurable sets, 16

non-measurable sets, 26

norm, 119, 129

nowhere dense, 35

odd function, 180

outer measure, 50

pointwise

convergence, 65, 66

limit, 65

positive part, 63

probability

density function, 61, 104

distribution, 60

measure, 39

space, 60, 104

theory, 59, 104

product

σ-algebra, 162

measure, 162, 168

projection theorem, 194

Radon-Nikodym

derivative, 103

theorem, 103

random variable, 60, 104

real measurable function, 56

restriction, 45

restriction of

the σ-algebra, 46

the measure space, 46

Riemann

integrable, 95, 119, 120

integral, 1, 2, 95

sum, 5

sample space, 60

seminorm, 118

separable, 43

simple

function, 71

measurable function, 63, 72

Sobolev space, 198

step function, 72

subadditivity, 20

supremum, 13, 64

symmetric difference, 79

tag set, 5

Tonelli's theorem, 169

topological space, 43

total variation, 146

translation invariance, 19

unbounded intervals, 11

upper

bound, 12

Darboux integral, 3

Darboux sum, 3

Vitali

cover, 151

covering lemma, 151

Young's inequality, 129

zero measure, 40