A Borel map into a separable Banach space is strongly measurable

Choose a countable dense sequence $(d_j{)}_{j\ge 1}$ in $B$. For each $n\ge 1$, assign $\omega$ to the first index $j$ for which $\mathrm{\parallel }f(\omega )-d_j\mathrm{\parallel }<2^{-n}$; such an index exists by density. The resulting sets $A_{n,j}$ form a measurable countable partition: each is the preimage of an open ball under $f$, with the finitely many earlier preimages removed.

Since the partial unions ${\bigcup }_{j\le N}{A}_{n,j}$ increase to $\mathrm{\Omega }$, [F3] gives an integer $N_n$ for which the omitted tail has measure at most $2^{-2n}$. Define $s_n=d_j$ on $A_{n,j}$ for $j\le N_n$, and $s_n=0$ on the omitted tail. Then $s_n$ is a finite-valued measurable simple function.

The tail events $T_n={\bigcup }_{j>N_n}{A}_{n,j}$ satisfy ${\sum }_{n}P(T_n)\le {\sum }_n{2}^{-2n}<\mathrm{\infty }$. By [F4], for every $N$, $P({\bigcup }_{n\ge N}{T}_n)\le {\sum }_{n\ge N}{2}^{-2n}$; continuity from above makes the probability of belonging to infinitely many $T_n$ equal to zero. Outside that null set, for all sufficiently large $n$ the point is not in $T_n$, and then $\mathrm{\parallel }{s}_n(\omega )-f(\omega )\mathrm{\parallel }<2^{-n}$. Thus $s_n\to f$ almost everywhere, which is strong measurability by [F2].