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.