| Title:
|
On coincidence of Pettis and McShane integrability (English) |
| Author:
|
Fabian, Marián |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
65 |
| Issue:
|
1 |
| Year:
|
2015 |
| Pages:
|
83-106 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
R. Deville and J. Rodríguez proved that, for every Hilbert generated space $X$, every Pettis integrable function $f\colon [0,1]\rightarrow X$ is McShane integrable. R. Avilés, G. Plebanek, and J. Rodríguez constructed a weakly compactly generated Banach space $X$ and a scalarly null (hence Pettis integrable) function from $[0,1]$ into $X$, which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from $[0,1]$ (mostly) into $C(K)$ spaces. We focus in more detail on the behavior of several concrete Eberlein (Corson) compact spaces $K$, that are not uniform Eberlein, with respect to the integrability of some natural scalarly negligible functions from $[0,1]$ into $C(K)$ in McShane sense. (English) |
| Keyword:
|
Pettis integral |
| Keyword:
|
McShane integral |
| Keyword:
|
MC-filling family |
| Keyword:
|
uniform Eberlein compact space |
| Keyword:
|
scalarly negligible function |
| Keyword:
|
Lebesgue injection |
| Keyword:
|
Hilbert generated space |
| Keyword:
|
strong Markuševič basis |
| Keyword:
|
adequate inflation |
| MSC:
|
46B26 |
| MSC:
|
46G10 |
| idZBL:
|
Zbl 06433722 |
| idMR:
|
MR3336026 |
| DOI:
|
10.1007/s10587-015-0161-x |
| . |
| Date available:
|
2015-04-01T12:22:38Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144214 |
| . |
| Reference:
|
[1] Argyros, S. A., Arvanitakis, A. D., Mercourakis, S. K.: Talagrand's $K_{\sigma\delta}$ problem.Topology Appl. 155 (2008), 1737-1755. Zbl 1158.46011, MR 2437025, 10.1016/j.topol.2008.05.014 |
| Reference:
|
[2] Argyros, S., Farmaki, V.: On the structure of weakly compact subsets of Hilbert spaces and applications to the geometry of Banach spaces.Trans. Am. Math. Soc. 289 (1985), 409-427. Zbl 0585.46010, MR 0779073, 10.1090/S0002-9947-1985-0779073-9 |
| Reference:
|
[3] Argyros, S., Mercourakis, S., Negrepontis, S.: Functional-analytic properties of Corson-compact spaces.Stud. Math. 89 (1988), 197-229. Zbl 0656.46014, MR 0956239, 10.4064/sm-89-3-197-229 |
| Reference:
|
[4] Avilés, A., Plebanek, G., Rodríguez, J.: The McShane integral in weakly compactly generated spaces.J. Funct. Anal. 259 (2010), 2776-2792. Zbl 1213.46037, MR 2719274, 10.1016/j.jfa.2010.08.007 |
| Reference:
|
[5] Benyamini, Y., Starbird, T.: Embedding weakly compact sets into Hilbert space.Isr. J. Math. 23 (1976), 137-141. Zbl 0325.46023, MR 0397372, 10.1007/BF02756793 |
| Reference:
|
[6] Číek, P., Fabian, M.: Adequate compacta which are Gul'ko or Talagrand.Serdica Math. J. 29 (2003), 55-64. MR 1981105 |
| Reference:
|
[7] Piazza, L. Di, Preiss, D.: When do McShane and Pettis integrals coincide? Ill.J. Math. 47 (2003), 1177-1187. MR 2036997, 10.1215/ijm/1258138098 |
| Reference:
|
[8] Deville, R., Rodríguez, J.: Integration in Hilbert generated Banach spaces.Isr. J. Math. 177 (2010), 285-306. MR 2684422, 10.1007/s11856-010-0047-4 |
| Reference:
|
[9] Fabian, M. J.: Gâteaux Differentiability of Convex Functions and Topology. Weak Asplund Spaces.Canadian Mathematical Society Series of Monographs and Advanced Texts Wiley, New York (1997). Zbl 0883.46011, MR 1461271 |
| Reference:
|
[10] Fabian, M., Godefroy, G., Montesinos, V., Zizler, V.: Inner characterizations of weakly compactly generated Banach spaces and their relatives.J. Math. Anal. Appl. 297 (2004), 419-455. Zbl 1063.46013, MR 2088670, 10.1016/j.jmaa.2004.02.015 |
| Reference:
|
[11] Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach Space Theory. The Basis for Linear and Nonlinear Analysis.CMS Books in Mathematics Springer, Berlin (2011). Zbl 1229.46001, MR 2766381 |
| Reference:
|
[12] Farmaki, V.: The structure of Eberlein, uniformly Eberlein and Talagrand compact spaces in $\Sigma(R^\Gamma)$.Fundam. Math. 128 (1987), 15-28. MR 0919286, 10.4064/fm-128-1-15-28 |
| Reference:
|
[13] Fremlin, D. H.: Measure Theory Vol. 4. Topological Measure Spaces Part I, II. Corrected second printing of the 2003 original.Torres Fremlin Colchester (2006). Zbl 1166.28001, MR 2462372 |
| Reference:
|
[14] Fremlin, D. H.: The generalized McShane integral.Ill. J. Math. 39 (1995), 39-67. Zbl 0810.28006, MR 1299648, 10.1215/ijm/1255986628 |
| Reference:
|
[15] Hájek, P., Montesinos, V., Vanderwerff, J., Zizler, V.: Biorthogonal Systems in Banach Spaces.CMS Books in Mathematics Springer, New York (2008). Zbl 1136.46001, MR 2359536 |
| Reference:
|
[16] Leiderman, A. G., Sokolov, G. A.: Adequate families of sets and Corson compacts.Commentat. Math. Univ. Carol. 25 (1984), 233-246. Zbl 0586.54022, MR 0768810 |
| Reference:
|
[17] Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I. Sequence Spaces.Ergebnisse der Mathematik und ihrer Grenzgebiete 92 Springer, Berlin (1977). Zbl 0362.46013, MR 0500056 |
| Reference:
|
[18] Lukeš, J., Malý, J.: Measure and Integral.Matfyzpress Praha (1995). Zbl 0888.28001 |
| Reference:
|
[19] Marciszewski, W.: On sequential convergence in weakly compact subsets of Banach spaces.Stud. Math. 112 (1995), 189-194. Zbl 0822.46004, MR 1311695, 10.4064/sm-112-2-189-194 |
| Reference:
|
[20] Martin, D. A., Solovay, R. M.: Internal Cohen extensions.Ann. Math. Logic 2 (1970), 143-178. Zbl 0222.02075, MR 0270904, 10.1016/0003-4843(70)90009-4 |
| Reference:
|
[21] Schwabik, Š., Ye, G.: Topics in Banach Space Integration.Series in Real Analysis 10 World Scientific, Hackensack (2005). Zbl 1088.28008, MR 2167754 |
| Reference:
|
[22] Talagrand, M.: Espaces de Banachs faiblement $\mathcal K$-analytiques.Ann. Math. 110 (1979), 407-438 French. MR 0554378, 10.2307/1971232 |
| . |
