Usage of "measurable" in measurable function

Question

Why are measurable functions called "measurable"? What exactly is being "measured"?

For measurable sets, I can intuitively understand that the measure of the set "measures" how large is the set.

Thanks for any help!

Answer 1

For a measurable function $f$, we can measure what is $\{f \leq 1\}$ or what is $\{0.5\leq f\leq 12\}$ etc.

Answer 2

Measuable functions preserve sigma-algebra structure i.e measurable sets.

Answer 3

1. What is so "real" about the real numbers?

2. Why is an open set called "open?"

3. Why are Cauchy sequences called "Cauchy?"

4. Why are Lipschitz functions called "Lipschitz?"

5. Why is $e$ called "e?"

6. Why are compact sets called "compact?"

7. Why are monotonic sequences called "monotonic?" [monotonic=speaking or uttered with an unchanging pitch or tone.]

8. Why are some metric spaces "separable?"

9. Why are some sets "barrelled?"

10. Why do we say some sets are "G-delta" and others are "F-sigma?"

I could go on, ... but I won't! The answer is always historical. If the word choice makes some sense in context then you are lucky. In English measurable="able to be measured" but mathematicians are under no obligation to choose language that makes sense in the vernacular.

Lebesgue probably took the language from the earlier Peano-Jordan theory of measure. Since he called, both sets and functions, measurable we are stuck with the language. Associate the term directly with the actual definition, not with any intuitive or suggestive meaning.

For the others in this list, some were named this way after famous mathematicians. The "open set" one is stranger: they first defined what a closed set was and declared that "open" meant "not closed" [like doors and windows]. Eventually the term acquired its current meaning which is very very different from "not closed" [to the dismay of some of my students].

The explanation for "G-delta" and "F-sigma" is curious: in French closed is ferme and sum is somme hence the $F_\sigma$. In German the words for open/intersection are Gebiet and Durchshnitt and that is why we use $G_\delta$.

In short if you feel like asking this kind of question, well ask away! But be prepared for a completely unsatisfying answer that amounts to nothing more that "that is what we always call them."

Answer 4

An integral is, at heart, a limit of sums of the form "value of the function times size of the region".

The idea behind the Lebesgue integral of a real function $f$ on a region $L\subset\mathbb R$, that addresses shortcomings in the Riemann integral, is to divide the codomain into small intervals $I_1,\ldots,I_n$ and take the limit of sums of the form $$\tag1 \sum_j t_j\,m(f^{-1}(I_j)), $$ where $t_j\in I_j$, $j=1,\ldots,n$. For this to make sense one needs to extend the notion of "length" from intervals (as used in the Riemann integral) to arbitrary sets. This turns out to be impossible if we want our measure $m$ to have reasonable properties. Lebesgue and others realized that one can still define $m$ if instead of considering every possible subset of $\mathbb R$ one restrict to a certain family of subsets, thus called measurable sets. The family of Lebesgue-measurable sets is big enough to cover almost any reasonable set that arises naturally.

The problem is that now, in $(1)$, $f^{-1}(I_j)$ cannot be any set, but rather it has to be a set where $m$ makes sense, i.e., a measurable set. So the Lebesgue integral will work only for functions where $f^{-1}(I)$ is measurable for every interval $I$. Such functions are call measurable, in the sense that their preimages of intervals are measurable.