Index 201

Lebesgue integral of a bounded

function, 51

Lebesgue measurable, 180

Lebesgue measurable function, 45

Lebesgue measurable set, 31, 182,

183, 189

Lebesgue measure, 31, 174, 190

Lebesgue outer measure, 175

Lebesgue simple, 42

limit, 145

limit in a normed space, 170

limit point, 158

linear function, 98

linear functional, 96, 106

linearity, 42

Littlewood’s three principles, 34, 72

lower bound, 142

Mean value theorem, 161

measurable function, 44, 45

measurable function (complex), 119

measurable partition, 41

measurable set, 31, 174, 182, 189

measurable subset of T , 126

measurable with respect to a

σ-algebra, 73

measure, 73

measure preserving transformation,

130

Minkowski inequality, 87

modulus, 149, 168

monotone, 146

monotone convergence theorem, 70

monotone decreasing, 146

monotone increasing, 146

monotonicity, 42, 181

movie, 139

natural numbers, 141

non-measurable set, 193

norm, 84, 163, 168

normed linear space, 164

nowhere dense, 39

null set, 28

open interval, 156

open set, 156

orbit, 129

orthogonal projection, 99, 107

orthonormal, 165

orthonormal family, 100, 106

outer measure, 175

parallelogram law, 164, 169

perfect, 39

perpendicular, 95, 168

Poincar´ e recurrence theorem, 133

pointwise, 56

positive definite, 88, 105, 163, 167

power set, 154

Pythagorean theorem, 95, 168

Radon-Nikodym derivative, 76

Radon-Nikodym theorem, 76

range, 143

rational numbers, 141

real numbers, 141

real part, 149

recurrent, 129, 132

regulated function, 9

regulated integral, 11

Riemann integrable, 18

Riemann integral, 18

sequence, 144

set difference, 32, 151

set inverse, 144

Sigma algebra, 29

simple, 42

simple functions, 41

square integrable, 84

square integrable complex

functions, 121

square summable, 148

standard Hermitian form, 167

step function, 5

subadditivity, 174

supremum, 142

surjection, 144

surjective, 143

symmetric difference, 35

translation invariance, 26, 31, 173,

178, 190

triangle inequality, 164, 169

uncountable, 152

uniform continuity, 160