| preface bibliography acknowledgments 1 introduction to rn a sets b countable sets c topology d compact sets e continuity f the distance function 2 lebesgue measure on rn a construction b properties of lebesgue measure c appendix: proof of p1 and p2 3 invariance of lebesgue measure a some linear algebra b translation and dilation c orthogonal matrices d the general matrix 4 some interesting sets .a a nonmeasurable set b a bevy of cantor sets c the lebesgue function d appendix: the modulus of continuity of the lebesgue functions 5 algebras of sets and measurable functions a algebras and a-algebras b borel sets c a measurable set which is not a borel set d measurable functions e simple functions 6 integration a nonnegative functions b general measurable functions c almost everywhere d integration over subsets of rn e generalization: measure spaces f some calculations g miscellany 7 lebesgue integral on rn a riemann integral b linear change of variables c approximation of functions in l1 d continuity of translation in l1 8 fubini's theorem for rn 9 the gamma function a definition and simple properties b generalization c the measure of balls d further properties of the gamma function e stirling's formula f the gamma function on r 10 lp spaces , a definition and basic inequalities b metric spaces and normed spaces c completeness of lp d the case p=∞ e relations between lp spaces f approximation by c∞c (rn) g miscellaneous problems ; h the case 0[p[1 11 products of abstract measures a products of 5-algebras b monotone classes c construction of the product measure d the fubini theorem e the generalized minkowski inequality 12 convolutions a formal properties b basic inequalities c approximate identities 13 fourier transform on rn a fourier transform of functions in l1 (rn) b the inversion theorem c the schwartz class d the fourier-plancherel transform e hilbert space f formal application to differential equations g bessel functions h special results for n= i i hermite polynomials 14 fourier series in one variable a periodic functions b trigonometric series c fourier coefficients d convergence of fourier series e summability of fourier series f a counterexample g parseval's identity h poisson summation formula i a special class of sine series 15 differentiation a the vitali covering theorem b the hardy-littlewood maximal function c lebesgue's differentiation theorem d the lebesgue set of a function e points of density f applications g the vitali covering theorem (again) h the besicovitch covering theorem i the lebesgue set of order p j change of variables k noninvertible mappings 16 differentiation for functions on r a monotone functions b jump functions c another theorem of fubini d bounded variation e absolute continuity f further discussion of absolute continuity g arc length h nowhere differentiable functions i convex functions index symbol index |
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