You might be interested in what Borel thought about this. Every modern
student learns that Lebesgue's measure is the completion of Borel's measure
and that this is rather obvious. Take the Cantor set of measure zero (i.e., Borel measure zero).
All subsets are Lebesgue measurable but not all subsets are Borel sets. It is clearly a "bad"
thing to have sets (and functions) around that your theory has to avoid, so of course Lebesgue's measure is
clearly more useful than Borel's.
But Borel didn't buy that. He was a bit of a constructionist. Not like you sometimes find
among people that don't believe in infinite sets or even infinite decimal expansions. He
didn't accept that any of these Lebesgue measurable sets that are not also Borel sets can
be constructed in any acceptable way. He had given a procedure (countable but transfinite) that
constructed all the Borel sets and Lebesgue had no demonstration that there were any other sets that
you could actually encounter.
Borel and Lebesgue were the best of friends---until they weren't. It was this issue that drove
them apart. Borel was a bit older and had supervised Lebesgue's dissertation. But he quite
resented the acclaim that Lebesgue was getting for his measure and his integral when the original
ideas were all due to Borel. If you fully believe that non Borel sets don't truly exist then
it appears Lebesgue has stolen the glory and with no justification. Priority disputes among mathematicians
are fairly rare, but they can be as bitter as such disputes in other fields.
I am not enough of an historian to tell much more of this story. (Of course that wouldn't stop me from
telling such stories in lectures.) But I would say that, at least formally, this dispute couldn't have been settled
until around 1914. That is when a young Russian mathematician (Suslin) showed that the projection of a two-dimensional Borel set
onto one-dimension need not be a Borel set, but did have to be Lebesgue measurable.
I hope that this would have settled the issue in Borel's mind but, if so, it did not restore their friendship.
But Borel might have enjoyed one aspect: Suslin made his discovery by finding a rather gross error in a paper
of Lebesgue's, a paper that claimed the projection of a Borel set would be a Borel set. The mistake Lebesgue
made was an embarrassingly simple one.