Darboux-integrability and uniform convergence (Q1766454)

A concept of Darboux-integrability for Banach space-valued functions defined on a basic space \((\Omega, {\mathcal D}, \mu)\) is introduced. In the finite-dimensional case for functions on \([a,b]\) this concept generalizes the Riemann integral, but in the infinite-dimensional case the Darboux-integral is more restrictive than the standard definition of the Riemann integral. In particular, the representation of a Darboux-integrable function as a uniform limit of simple functions remains valid in infinite-dimensional spaces.