For a Banach-valued random mapping $f:\Omega\rightarrow X$, there are three kind of measurability: strong measurability (can be approximated by sequence of simple functions, measurability (the preimage of each measurable set is measurable) and weak measurability (the composition with any element of the dual space is a real-valued measurable function in the usual sense); see the book of Neerven https://fa.ewi.tudelft.nl/~neerven/publications/notes/ISEM.pdf. The well-known Pettis theorem (also see the above reference) tells us that the strong measurability implies both measurability and weak measurability. But what is the relationship between measurability and weak measurability? Seems there is no book discusses this issue.