Conditional expectations and weak and strong compactness in spaces of Bochner integrable functions

James K. Brooks and Nicolae Dinculeanu

Journal of Multivariate Analysis, 1979, vol. 9, issue 3, 420-427

Abstract: In [6, theorem IV.8.18], relatively norm compact sets K in Lp([mu]) are characterized by means of strong convergence of conditional expectations, E[pi]f --> f in Lp([mu]), uniformly for f [set membership, variant] K, where (E[pi]) is the family of conditional expectations corresponding to the net of all finite measurable partitions. In this paper we extend the above result in several ways: we consider nets of not necessarily finite partitions; we consider spaces of vector valued pth power Bochner integrable functions (and spaces M([Sigma], E) of vector valued measures with finite variation); we characterize relatively strong compact sets K in by means of uniform strong convergence E[pi]f --> f, as well as relatively weak compact sets Kby means of uniform weak convergence E[pi]f --> f. Previously, in [4], uniform strong convergence (together with some other conditions) was proved to be sufficient (but not necessary) for relative weak compactness.

Keywords: conditional; expectations; compactness; Bochner; integrable; functions (search for similar items in EconPapers)
Date: 1979
References: Add references at CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/0047-259X(79)90099-X
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:jmvana:v:9:y:1979:i:3:p:420-427

Ordering information: This journal article can be ordered from
http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

Access Statistics for this article

Journal of Multivariate Analysis is currently edited by de Leeuw, J.

More articles in Journal of Multivariate Analysis from Elsevier
Bibliographic data for series maintained by Catherine Liu ().