Advantage of Lebesgue sigma-algebra over Borel? What it says on the tin. Using the Borel $\sigma$-algebra on the reals instead of the Lebesgue $\sigma$-algebra has the advantage that it allows a broader class of measures, many of which are quite natural: For example the "uniform" measure on the Cantor set is defined on the Borel $\sigma$-algebra, but cannot be defined on the Lebesgue algebra. So why don't we just use the Borel $\sigma$-algebra for everything? What advantage does the Lebesgue $\sigma$-algebra have?
I mean, it has more measurable sets, but sets that are Lebesgue-measurable but not Borel-measurable (or for that matter, sets that are not Borel-measurable, period) are extremely pathological, not explicitly constructible, and (as far as I can tell) never show up naturally. And it's complete, but I have no idea what makes that a useful property.
 A: 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.  
