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Let $E$ be a Banach space. Define step map $f:[a,b]\to E$ as a map for which there is a partition $a=a_0, a_1, \ldots, a_n = b$ and elements $v_1,\ldots, v_n\in E$ such that $f(t)=v_i$ for $t\in(a_{i-1},a_i)$. $a,b,E$ are fixed.

What is the closure of the space of step maps with respect to sup norm?

Motivation: we can define integral $\int_a^b f$ as sum for step maps, this integral is continuous function, so we can extend integral to the closure of stepmaps. This extension is well defined and unique. So, it's interesting what is the full class of integrable maps.

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  • $\begingroup$ I would conjecture that the uniform closure is the collection of all bounded integrable functions. $\endgroup$
    – Joel
    Jun 11, 2014 at 15:42

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