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.