Let $B\_{\infty}(\Omega)$ be the space of bounded measurable functions on the measurable space $\Omega$. For a given Banach space $X$, let us denote $B\_{\infty}(\Omega,X)$ by the set of all bounded measurable functions $f:\Omega\to X$ (meaning $f^{-1}(O)$ is measurable for every open $O$ in $X$). 



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> Is the space 
>  $B\_{\infty}(\Omega,X)$ just the invective tensor product of $X$ and $B\_{\infty}(\Omega)$?
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A formula for vector valued measurable functions
For $\Omega=\mathbb N$ with the disrete $\sigma$-algebra measurability is no condition and $$\ell^\infty(\mathbb N)\hat\otimes\_\varepsilon X \cong C(\beta\mathbb N)\hat\otimes\_\varepsilon X\cong C(\beta\mathbb N,X)$$ is (via restriction to $\mathbb N$ isomorphic to) the space of *pre-compact* sequences in $X$.


