TITLE: Why don't we define a measurable function to be inverse image of measurable set is measurable
QUESTION [5 upvotes]: In royden's book, he defines a measurable function satisfying that inverse image of $[a, + \infty )$ is measurable. It seems to be quite restrictive, and I understand that my definition in the topic is more general. What are the drawbacks of my definition?

REPLY [2 votes]: Let me rephrase the question a little bit:
Why a function $f: R\to R$ is defined measurable only in the sense of $f: (R, L)\to (R, B),$ where $L$ is Lebesgue measure, and $B$ is Borel measure. In other words, what is the point of non-symmetry here? Why not define a measurable function in the sense of $f: (R, L)\to(R, L)$.
The answer is that to introduce integral, measurable functions in the former sense is sufficient. While measurable in the latter sense is actually more restrictive. And there would be  no extra bonus for that restriction.
The last point is only my own opinion. it is nearly impossible that one encounters a function which is measurable in the former sense but not measurable in the latter sense, unless of course someone construct  a counter example deliberately. That function must be very strange, it is not likely you need to deal with one in real life.