This is an announcement for the paper "Lusin's Theorem and Bochner integration" by Peter A. Loeb and Erik Talvila.
Abstract: It is shown that the approximating functions used to define the Bochner integral can be formed using geometrically nice sets, such as balls, from a differentiation basis. Moreover, every appropriate sum of this form will be within a preassigned $\varepsilon$ of the integral, with the sum for the local errors also less than $\varepsilon$. All of this follows from the ubiquity of Lebesgue points, which is a consequence of Lusin's theorem, for which a simple proof is included in the discussion.
Archive classification: Classical Analysis and ODEs; Functional Analysis
Mathematics Subject Classification: 28A20, 28B05; 26A39
Remarks: To appear in Scientiae Mathematicae Japonicae
The source file(s), bochnerbox.tex: 34366 bytes, is(are) stored in gzipped form as 0406370.gz with size 11kb. The corresponding postcript file has gzipped size 52kb.
Submitted from: etalvila@math.ualberta.ca
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/math.CA/0406370
or
http://arXiv.org/abs/math.CA/0406370
or by email in unzipped form by transmitting an empty message with subject line
uget 0406370
or in gzipped form by using subject line
get 0406370
to: math@arXiv.org.