Author | Kannan, R. author |
---|---|
Title | Advanced Analysis [electronic resource] : on the Real Line / by R. Kannan, Carole King Krueger |
Imprint | New York, NY : Springer New York, 1996 |
Connect to | http://dx.doi.org/10.1007/978-1-4613-8474-8 |
Descript | X, 260 p. online resource |
0 Preliminaries -- 0.1 Lebesgue Measure -- 0.2 The Lebesgue Integral -- 0.3 Vitali Covering Theorem -- 0.4 Baire Category Theorem and Baire Class Functions -- 1 Monotone Functions -- 1.1 Continuity Properties -- 1.2 Differentiability Properties -- 1.3 Reconstruction of f from f? -- 1.4 Series of Monotone Functions -- Exercises -- 2 Density and Approximate Continuity -- 2.1 Preliminaries and Definitions -- 2.2 The Lebesgue Density Theorem -- 2.3 Approximate Continuity -- 2.4 Approximate Continuity and Integrability -- 2.5 Further Results on Approximate Continuity -- 2.6 Sierpinskiโs Theorem -- 2.7 The Darboux Property and the Density Topology -- Exercises -- 3 Dini Derivatives -- 3.1 Preliminaries and Definitions -- 3.2 Simple Properties of Derivatives -- 3.3 Ruziewiczโs Example -- 3.4 Further Properties of Derivatives -- 3.5 The Denjoy-Saks-Young Theorem -- 3.6 Measurability of Dini Derivatives -- 3.7 Dini Derivatives and Convex Functions -- Exercises -- 4 Approximate Derivatives -- 4.1 Definitions -- 4.2 Measurability of Approximate Derivatives -- 4.3 Analogue of the Denjoy-Saks-Young Theorem -- 4.4 Category Results for Approximate Derivatives -- 4.5 Other Properties of Approximate Derivatives -- Exercises -- 5 Additional Results on Derivatives -- 5.1 Derivatives -- 5.2 Derivates -- 5.3 Approximate Derivatives -- 5.4 The Denjoy Property -- 5.5 Metrically Dense -- 6 Bounded Variation -- 6.1 Bounded Variation of Finite Intervals -- 6.2 Stieltjes Integral -- 6.3 The Space BV[a,b] -- BVloc and L1loc -- 6.5 Additional Remarks on Fubiniโs Theorem -- Exercises -- 7 Absolute Continuity -- 7.1 Absolute Continuity -- 7.2 Rectifiable Curves -- Exercises -- 8 Cantor Sets and Singular Functions -- 8.1 The Cantor Ternary Set and Function -- 8.2 Hausdorff Measure -- 8.3 Generalized Cantor SetsโPart I -- 8.4 Generalized Cantor SetsโPart II -- 8.5 Cantor-like Sets -- 8.6 Strictly Increasing Singular Functions -- Exercises -- 9 Spaces of BV and AC Functions -- 9.1 Convergence in Variation -- 9.2 Convergence in Length -- 9.3 Norms on AC -- 9.4 Norms on BV -- 10 Metric Separability -- Exercises