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Riemann step function approximation of Bochner integrable functions

Author: M. A. Freedman
Journal: Proc. Amer. Math. Soc. 96 (1986), 605-613
MSC: Primary 28B05; Secondary 34A60
MathSciNet review: 826489
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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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