Gerhard Grimeisen
Approximation of the Bochner integral by means of Riemann sums
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 54, 3-37 (2000)

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

Abstract: The natural way to approximate the integral (of some kind) of a
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 $f$.
Keywords: Vector-valued set functions, measures and integrals, Vector-valued measures and integration, Set functions and measures on topological spaces (regularity of measures, etc.)