I'm looking for a counter exemple (with proof please) of a mesurable map $$ f : X \rightarrow E $$ with $X$ a measurable space et $E$ a $\mathbb{R}$ Banach space. Such as $f$ is not the limit of sequence of simple functions.
Thanks and regards.
2026-03-28 15:26:54.1774711614
A measurable map in Banach space and simple functions.
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Let $E$ be any non-separable Banach space, $X=E$ with the Borel sigma algebra. Let $f$ be the identity map of $X$. Then $f$ is Borel - Borel measurable. If there is a sequence $(f_n)$ of simple functions converging pointwise to $f$ then the range of $f$ would be contained in the closed subspace generated by countable set (namely the union of the ranges of $f_n$'s). But the range is $E$ which is not separable.