analysis - 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?