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 Section 1.2.1 in the book of Neerven https://fa.ewi.tudelft.nl/~neerven/publications/notes/ISEM.pdf. The well-known Pettis theorem (see Theorem 1.5 in Section 1.2.1 of the above reference book) tells us that the strong measurability implies both measurability and weak measurability ($f$ is strongly measurable if it is measurable (or weakly measurable) and separably valued). But what is the relationship between measurability and weak measurability? Seems there is no book discusses this issue (If $X$ is separable, these three kind of measurability are equivalent due to the Pettis theorem, but what about for the general (non-separable) Banach space $X$?).

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