Be $(\Omega, \mathcal{A}, P)$ and $E$ a probability space and a Banach space respectively.
This paper of G.A. Edgar contains a proof that, for a function $X: \Omega \rightarrow E$, being weakly measurable is equivalent to being a $\big(\mathcal{A}, {\mathcal{Ba}}(E,\sigma(E, E^{\,\vee}))\big)$-measurable function. ${\mathcal{Ba}}(E,\sigma(E, E^{\,\vee}))$ is the Baire $\sigma$-algebra w.r.t. the weak topology on $E$.
If $X:\Omega\to E$ is a strongly/Bochner measurable function this is equivalent to being weakly measurable and separably valued or equivalent to being $(\mathcal{A}, \mathcal{B}(E))$-measurable and separably valued.
Question: If $X: \Omega \rightarrow E$ is a strongly/Bochner measurable function, does there exist a characterization of it solely as a random element i.e. an $(\mathcal{A}, \mathcal{E})$-measurable function for some $\sigma$-algebra $\mathcal{E}$ ? (Looks like the property 'separably valued' needs to be encoded into such a $\sigma$-algebra)
