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Riemann step function approximation of Bochner integrable functions. (English) Zbl 0595.28013
Given a Banach space $$(X,\| \cdot \|)$$, consider the space $$L^ 1(0,T;X)$$ of all X-valued, Bochner integrable functions f defined a.e. on the interval $$[0,T],$$ and having norm $$\| f\|_{L^ 1(0,T;X)}=\int^{T}_{0}\| f(t)\| dt.$$ The author shows 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.
Reviewer: O.Lipovan
##### MSC:
 28B05 Vector-valued set functions, measures and integrals
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##### References:
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