Riemann step function approximation of Bochner integrable functions

Freedman, M. A.

doi:10.1090/S0002-9939-1986-0826489-4

Riemann step function approximation of Bochner integrable functions
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Proc. Amer. Math. Soc. 96 (1986), 605-613 Request permission

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.

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