Let $(\Omega, \Sigma)$ be a measurable space and $X$ a Banach space. Let $f: \Omega \rightarrow X$.
- $f$ is called measurable if every the preimage of every Borel set in $X$ is an element of $\Sigma$.
- $f$ is called strongly measurable if $f$ is the pointwise limit of a sequence of simple functions.
It is known that strongly measurable and measurable are equivalent when $X$ is separable. For this reason, the notion of strong measurability is only relevant when dealing with Bochner integration in full generality. What is an example of a function $f$ taking values in a non-separable Banach space $X$ such that $f$ is measurable but not strongly measurable?
Best Answer
If you take $(\Omega,\Sigma)$ as $(X,\mathcal{B}(X))$, where $X$ is a non-separable Banach space, then the identity function $I:X\longmapsto X$ is not strongly measurable (but is continuous so is Borel measurable).