I wonder if we can characterize weak measurability of a function taking values in a Banach space using sequence of step functions (functions that have finite range) just like how we define strong measurability? 


More specifically, a function $f:\Omega\mapsto X$ defined on a measure space $(\Omega,\Sigma,\mu)$ and taking values in a Banach space $X$ is *strongly measurable* if there exists a sequence of step functions $\{ \phi\_n \}$ such that $\phi\_n\rightarrow f$ in norm a.e.. Could we analogously say that $f$ is *weakly measurable* iff there exists a sequence of step functions $\{ \phi\_n \}$ such that $\phi\_n\rightarrow f$ weakly a.e.? One direction is obviously true, but I can't figure out the other direction.


For reference, here is the definition of weak measurability: A function $f:\Omega\mapsto X$ is *weakly measurable* if $\langle f(\omega), x \rangle$ is measurable for each $x\in X'$, the norm dual of $X$.



Characterization of Weakly measurable functions
If there is a sequence of step functions such that $\phi\_n\to f$ weakly a.e., then $f$ is almost separably valued. But if it is weakly measurable and almost separably valued, it is strongly measurable.


