The definition of a measurable function From my understanding there are two main ways of defining a measurable function from a measure space to a Banach space (whose base field contains the real numbers): a function whose preimage maps Borel sets to measurable sets or a function that is a pointwise limit of simple functions. 
I believe that these two definitions are equivalent when the Banach space is a finite dimensional space over the real numbers. But if the Banach space is not finite dimensional, then being a pointwise limit of simple functions becomes a stronger condition. How does one (or where can one find) a proof that if a function is a pointwise limit of simple functions, then its preimage maps Borel sets to measurable sets?
Also what it the point of (ever) defining measurable functions to be functions whose preimage maps Borel sets to measurable sets? Is it just because the one implication is easier than the other?
 A: If a function $f:\Omega\to X$ is the pointwise limit of a sequence $\langle f_n\rangle$ of simple functions, then $$\mathrm{cl}\big(f(\Omega)\big)\subseteq \mathrm{cl}\Big(\bigcup_n f_n(\Omega)\Big)$$ and 
$\bigcup_n f_n(\Omega)$ is countable. It follows that $f$ must have separable range. If $X=\Omega$ is a non-separable Banach space, then the preimage of a Borel set is itself and therefore measurable when we endow $\Omega$ with the Borel $\sigma$-algebra. But the identity is not the pointwise limit of simple functions.
Now let again $f:\Omega\to X$ be the pointwise limit of a sequence $\langle f_n\rangle$ of simple functions. To show that the preimage of a Borel set under $f$ is measurable, it suffices to show that the preimage of an open set under $f$ is measurable, since the open sets generate the Borel sets. So let $O\subseteq X$ be open. Then
$$f^{-1}(O)=\{\omega\in\Omega:f(\omega)\in O\}=\{\omega\in\Omega:\lim_{n\to\infty}f_n(\omega)\in O\}$$
$$=\{\omega\in\Omega:\text{there is some }N\text{ such that}f_n(\omega)\in O\text{ for }n\geq N\}$$
$$=\Big\{\omega\in\Omega:\text{there is some }N\text{ such that }\omega\in\bigcap_{n=N}^\infty f^{-1}_n(O)\Big\}$$
$$=\bigcup_{N=1}^\infty\bigcap_{n=N}^\infty f^{-1}_n(O)$$
and that set is measurable as the countable union of countable intersections of measurable sets. 
The reason for the preimage definition becomes obvious when one does somemeasure theoretic probability theorem. If I want to calculate the probability that some random variable takes a value in some interval, I need to be able to assign probabilities to preimages of intervals and hence Borel sets. This makes even sense for random variables with values in more abstract spaces.
