Riemann Step Function Approximation of Bochner Integrable Functions

Abstract

Let ${L^1}(0,T;X)$ denote the space of all Bochner integrable functions $f$ which map the interval $[0,T]$ into the Banach space $X$. Then we show that $f$ is the uniform limit in the ${L^1}$-norm of its Riemann step function approximations along nearly every sequence of partitions of $[0,T]$ with mesh size approaching zero.