# 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.

• If you have choice, I think it is not so hard to come up with an absolutely continuous function whose derivative fails to exist on a Lebesgue measurable, Borel nonmeasurable, measure zero set. Thus the fundamental theorem of calculus does not hold the way we would like in the Borel setting. By the way, some sets that are Lebesgue measurable and Borel nonmeasurable are explicitly constructible, it's just that the nonconstructive proof of their existence (using a Lebesgue nonmeasurable set as a starting point) is easier. – Ian Feb 3 '16 at 2:38
• Lebesgue is complete. – Aloizio Macedo Feb 3 '16 at 2:38
• The Borel $\sigma$-algebra doesn't even include all Jordan measurable sets, so you would end up with something weaker than the Riemann integral. – user310283 Feb 3 '16 at 2:41
• @user310283 It's not strictly weaker, since the inclusion doesn't go either way. Still a good point. – Ian Feb 3 '16 at 2:57
• @Ian In fact for a completely arbitrary function $F$ the set of points at which the derivative $F'$ exists is a Borel set. (Due, I believe, to O. Hájek (1957).) So one does have to go in some other direction to find natural examples of non Borel sets that arise in analysis. – B. S. Thomson Feb 3 '16 at 20:18