Motivation behind the definition of measurable functions.

Question

In measure theory,we work with non-pathological sets and functions called measurable sets and functions.The definition of measurable function $f:(X,\mathcal S)\to \mathbb R$ is defined as a function such that $f^{-1}(B)$ is $\mathcal S$-measurable for any $B\in \mathcal B_\mathbb R$,the Borel $\sigma$-algebra.More generally a measurable function between two measurable spaces $(X,\mathcal S)$ and $(Y,\mathcal F)$ is defined to be a function $f:X\to Y$ such that $E\subset Y$ is $\mathcal F$-measurable $\implies$ $f^{-1}(E)$ is $\mathcal S$-measurable.My question is why these functions are well behaved and non-pathological.How did mathematicians come up with such a strange criterion?What is the motivation behind the definition of measurable functions?Are they nice in the sense that they can be approximated by a sequence of simple functions?

Answer 1

We very often care about maps that preserve the structure of the spaces. For example, for topological spaces, a continuous map is one which preserves the openness of a set (see this post for intuition and for why we care). This is analogous to measurable maps (in fact, measurable maps are continuous). They preserve the measurability of a set. The standard example for a non-measurable function is the indicator of a set $A \subset X$, where $A \not \in \mathcal{S}$ (i.e. $A$ is not a measurable set). Then, the pre-image of $\{1\}$ is $A$, which is by definition not in $\mathcal{S}$. Now, such sets $A$ are hard to construct and are rather unintuitive, but the point is that measurable functions allow you to use the "information" in the domain to conclude something about the range. In this case, information about the measurable set $\{1\}$ cannot be traced back to any information in the domain, and we wouldn't be able to do much with it (such as integrating).

The fact that non-negative measurable maps can be approximated by an increasing sequence of simple functions is a consequence of measurability. Note that simple functions are defined to be nice. So although it is true that measurable functions can be approximated by a sequence of simple functions, in my opinion it doesn't tell you much about measurability.

In summary, measurable maps are maps that preserve the structure of measurable spaces. I hope this helps.