Lusin's Theorem and Bochner Integration

Details Lusin's Theorem and Bochner Integration Lusin's Theorem and Bochner Integration

2004-06-18

arxiv.org/abs/math/0406370v1

Mathematics

Classical Analysis and ODEs

To appear in Scientiae Mathematicae Japonicae

Scientific paper

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 $\epsilon$ of the integral, with the sum for the local errors also less than $\epsilon$. 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.

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