In an attempt to extend this result to Banach spaces, I come across the existence of some Borel sets.
Let $(\Omega, \mathcal F, \mu)$ be a $\sigma$-finite complete measure space. Let $(E,|\cdot|)$ be a Banach space and $\mathcal B_E$ its Borel $\sigma$-algebra. Let $f:\Omega \to E$ be $\mu$-measurable. By this theorem, $f$ is $\mathcal F$-$\mathcal B_E$ measurable. Let $\nu:= f_\sharp \mu$ be the push-forward measure. This means $\nu$ is a Borel probability measure on $E$.
Is there a Borel set $B \in \mathcal B_E$ such that $B \subset \operatorname{im} f := f(\Omega)$ and $\nu (B) = 1$?
Thank you for your elaboration!
Update: I have found a closely related question to the one above: Is there a $\mu$-measurable function $g:\Omega \to E$ such that $f=g$ $\mu$-a.e. such that $g(\Omega) \in \mathcal B_E$?