TITLE: Definition of Lebesgue measurable functions- Why Borel sets?
QUESTION [6 upvotes]: I write $\mathcal{M}_{Bor}, \mathcal{M}_{Leb}$ for the Borel/ Lebesgue $\sigma$- algebras on $\mathbb{R}$.
Let $A \in \mathcal{M}_{Leb}$. Let $f: A\to \mathbb{R}$. Then $f$ is Lebesgue- measurable iff $ \forall B \in \mathcal{M}_{Bor}: f^{-1}(B) \in \mathcal{M}_{Leb}$.
What goes wrong if we instead require $ \forall E \in \mathcal{M}_{Leb}: f^{-1}(E) \in \mathcal{M}_{Leb}$, for $f$ to be measurable? Superficially this definition seems more natural, so why isn't it used?

REPLY [2 votes]: It has do with the construction of the specific underlying measure (in your case Lebesgue measure).
In the beginning, you start with intervals (step functions) or continuous functions. Then, through some machinery (Daniell's approach or Lebesgue-Charatheodoty approach) you build integration theory (you extend you notion of integral or measure) Then you realize there is much more stuff that what you started with (Lebesgue sets and Lebesgue measurable functions)
Step functions and continuous functions are akin to  Borel sets (countable opertions with closed/open intervals, preimages  of intervals by continuous functions, etc).
If you are to do the same with a different measure $\mu$, that is build integration with respect another measure staring with step functions or continuous functions,  then you may get a different extension (different measurable and measurable functions). Which extension is the most natural?  In any case, both extensions contain the Borel sets and any other additional set (or function) we obtained can be approximated by Borel sets.
You can also ask about the intersection of all possible extensions one gets starting with step functions (or continuous function). That intersection contains the Borel sets properly, but is much smaller that the collection of Lebesgue sets (of functions). That intersection, the universal $\sigma$-algebra, is the natural extension and  it is important in the study of stochastic processes and in Measure theory.
Things become more complicated, but the same principles apply, when you start doing integration in topological spaces.