**MSC:**- 28B05 Vector-valued set functions, measures and integrals, See also {46G10}
- 46G10 Vector-valued measures and integration, See also {28Bxx,
- 28C15 Set functions and measures on topological spaces

function $f$ on a measure space $T$ into a Banach space $E$

via the values of $f$ is - in the author's opinion - that by Riemann sums. The Bochner

integral does not offer this possible way of approximation in an

immediate way. The goal of this paper is to investigate in which

sense the desired approximation is nevertheless possible for the

Bochner integral of $f$.