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
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