Summary and Info
Chapter I: Measures Definition of measure 3; regular measure 4; outer measure 6; measurable set 8; Hahn extension theorem 10; monotone families of sets 12; completion of measures 14; Borel sets and construc- tion of Lebesgue measure in the real line 15; the Cantor set 16; a non -measurable set 16. Chapter II: Integration Measure spaces and measurable functions 19; definition of the integral for non -negative functions 22; the integral as a measure 23; linearity of the integral 25; monotone convergence theorem 28; Fatou's lemma 30; the integral for real-valued functions 32; the integral for complex -val ued functions 33; dominated convergence theorem 34; bounded convergence theorem 35; Egoroff's theorem 36; convergence in measure 37; L -convergence 39; the Lebesgue p and Riemann integrals 41. Chapter III: The Theorems of Fubini Definition of :r x 'Y 45; simple functions 49; interchange of integration for non -negative functions 51; interchange of integra- tion for Ll functions 51' completion of X x y 52; a theorem on change of variables in integration 54. Chapter IV: Representations of Measures Definition of III I 58; Jordan decomposition theorem 60; Hahn decomposition theorem 62; the integral for complex measures 64; functions of bounded variation 64; the Cantor function 66; abso- lutely continuous measures 67; Radon-Nikodyn theorem 69; the Radon-Nikodyn derivative 73; mutually singular measures 73; Lebesgue decomposition theorem 74; Chapter V: The Lebesgue Spaces Holder's inequality 79; completeness of L 81; functions on p regular measure spaces 84; continuity of translation in L norm 85; p continuous linear functionals 86; weak and strong convergence of functionals 88; the dual spaces of Lp 91; the dual space of Co(S) 95. Chapter VI: Differentiation Derivative of a measurel07; Vitali covering theorem 108; upper and lower derivates 110; regularity of finite Borel measures 111; existence of Dfl (x) a.e. 114; convergence of measures 116; points of density and dispersion 117; differentiation of integrals 118; dif- ferentiation of singular measures 119; the Lebesgue set 120; inte- gration by parts 122. Chapter VII: Fourier Series Orthogonal systems of functions 123; definition of Fourier series and Fourier coefficients 125; Bessel's inequality 126; Riesz- Fischer theorem 127; complete systems 128; completeness of ex- ponential and trigonometric systems 129; Poisson kernel 132; posi- tive kernels 136; Riemann -Lebesgue theorem 139; the kernel D (u) n 141; Dirichlet- Jordan theorem 142; integration of Fourier series 147; Abel summability of Fourier series 149; (C, 1) summability of Fourier series 149; (C, 1) summability of Fourier series on L 1 151; a con- tinuous function whose Fourier series diverges at 0 153; uniform boundedness theorem 156; L convergence of Fourier series 157; p M. Riesz interpolation theorem 163; Hausdorff- Young theorem 166.