Index

C complex numbers, 141

∆ symmetric difference, 35

imaginary part, 149

M Lebesgue measurable sets, 31

M(I) measurable subsets of I, 31

N natural numbers, 141

Q rational numbers, 141

R real numbers, 141

real part, 149

Z integers, 141

L2[a,

b], square integrable

functions, 84

L2[a,

C

b], square integrable complex

functions, 121

2, 148

absolutely continuous, 74

M movie, 139

µ Lebesgue measure, 31, 182

ν-almost all, 130

⊥ perpendicular, 95

σ-algebra, 29, 37, 76

σ-algebra generated by a family of

sets, 30

absolute continuity, 65

absolute convergence, 95, 107, 147,

150

absolutely continuous, 74

absolutely continuous measure, 74

additivity, 54

algebra of functions, 114

almost all, 58

almost everywhere, 58

Axiom of choice, 155, 193, 194

Bessel’s inequality, 102, 109

bijection, 144

bijective, 143

bilinearity, 88, 163

Birkhoff ergodic theorem, 137

Borel σ-algebra, 30

Borel sets, 30

bounded convergence theorem, 56,

58

Cantor middle third set, 38

Carleson’s theorem, 117, 125

Cartesian product, 143, 153

Cauchy sequence, 91, 147, 150, 171

Cauchy-Schwarz inequality, 86, 96,

163, 168

characteristic function, 41

classical Fourier coeﬃcients, 113

closed, 156

closed interval, 156

closed subspace, 99

codomain, 143

commutativity, 88, 163

compact, 158

complement, 151

199