Panchapagesan T.V. The Bartle-Dunford-Schwartz integral: integration with respect to a sigma-additive vector measure / ed. by Wojtaszczyk P. et al.; Instytut matematyczny PAN. - 2008. - Basel: Birkhäuser Verlag AG, 2008. - xv, 301 p. - (Monografie matematyczne. New series; 69). - Bibliogr.: p.287-292; Ind.: p.297-301. - ISBN 978-3-7643-8601-6  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / Contents

Preface ........................................................ ix 1 Preliminaries 1.1 Banach space-valued measures ............................ 1 1.2 lcHs-valued measures ................................... 11 2 Basic Properties of the Bartle-Dunford-Schwartz Integral 2.1 (KL) m-integrability ................................... 17 2.2 (BDS) m-integrability .................................. 26 3 p-spaces, 1 ≤ p ≤ ∞ 3.1 The seminorms m•p(•,T) on p(m), 1 ≤ p < ∞ ........... 33 3.2 Completeness of p(m) and p(m), 1 ≤ p < ∞, ∞(m) ... 42 3.3 Characterizations of p(m), 1 ≤ p < ∞ ................. 47 3.4 Other convergence theorems for p(m), 1 ≤ p < ∞ ........ 54 3.5 Relations between the spaces p(m) ..................... 63 4 Integration With Respect to IcHs-valued Measures 4.1 (KL) m-integrability (m lcHs-valued) ................... 65 4.2 (BDS) m-integrability (m lcHs-valued) .................. 78 4.3 The locally convex spaces p(m), p(σ(Ρ),m), p(m) and p(σ(Ρ),m), 1 ≤ p < ∞ ...................... 85 4.4 Completeness of p(m), p(m), p(σ(Ρ),m) and p(σ(Ρ),m), for suitable X ............................ 91 4.5 Characterizations of p-spaces, convergence theorems and relations between p-spaces .............. 101 4.6 Separability of p(m) and p(σ(Ρ),m), 1 ≤ p < ∞, m IcHs-valued ......................................... 109 5 Applications to Integration in Locally Compact Hausdorff Spaces - Part I 5.1 Generalizations of the Vitali-Carathéodory Integrability Criterion Theorem ....................... 117 5.2 The Baire version of the Dieudonné-Grothendieck theorem and its vector-valued generalizations ......... 121 5.3 Weakly compact bounded Radon operators and prolongable Radon operators ........................... 138 6 Applications to Integration in Locally Compact Hausdorff Spaces - Part II 6.1 Generalized Lusin's Theorem and its variants .......... 153 6.2 Lusin measurability of functions and sets ............. 159 6.3 Theorems of integrability criteria .................... 165 6.4 Additional convergence theorems ....................... 187 6.5 Duals of 1(m) and 1(n) .............................. 202 7 Complements to the Thomas Theory 7.1 Integration of complex functions with respect to a Radon operator ................................... 213 7.2 Integration with respect to a weakly compact bounded Radon operator ........................................ 221 7.3 Integration with respect to a prolongable Radon operator .............................................. 233 7.4 Baire versions of Proposition 4.8 and Theorem 4.9 of [T] ................................................ 241 7.5 Weakly compact and prolongable Radon vector measures .............................................. 251 7.6 Relation between p(u) and p(mu), u a weakly compact bounded Radon operator or a prolongable Radon operator ........................................ 275 Bibliography .................................................. 287 Acknowledgment ................................................ 293 List of Symbols ............................................... 295 Index ......................................................... 297