Lusin's Theorem and Bochner Integration

TL;DR

This paper demonstrates that Bochner integrals can be approximated using geometrically simple sets like balls, leveraging Lebesgue points and Lusin's theorem for precise error control.

Contribution

It introduces a method to approximate Bochner integrals with geometrically nice sets, simplifying the process and providing explicit error bounds.

Findings

Approximation functions can be formed using balls from a differentiation basis.

Every sum of this form approximates the integral within a specified epsilon.

The approach relies on Lebesgue points and Lusin's theorem for error control.

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 $\epsilon$ 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.