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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
28B05 Vector-valued set functions, measures and integrals
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