A friend and I were having a discussion about Lebesgue measure. I attempted to be profound by making the following points:
- Analytic geometry has been a fantastic tool, but the concept of
representing a continuous "object" as a collection of points is
inherently contrived (with a negative connotation). Immediately we
run into the paradox that a point has no volume, and yet a collection
of many points has volume. - The notion of Lebesgue measure attempts to resolve this essential
tension by allowing only countable additivity of the measure. But it
does so only by disallowing certain operations (uncountable sums)
that intuitively seem reasonable. As such, it is an indispensable
tool, but it remains contrived on some level.
My friend countered by saying that uncountable additivity doesn't really make sense anyway, since any uncountable sum that converges must have co-countably many terms zero.
I would say I am still on the fence about this discussion. He makes a good point, but after all, it is exactly the addition of uncountably many zeros that I am concerned with, so the notion that co-countably many of the terms must be zero may not be a decisive objection.
Best Answer
Well it may be a drawback, but, taking the pragmatic route, what's the alternative? If we want a useful measure that is translation invariant and gives the length for intervals, then there will be non-measurable Vitali sets. Lebesgue's still seems to be the best measure we can get.
Maybe you could take some sort of nonstandard analysis approach and declare the measure of a point to be an uncountably small infinitesimal, so that uncountably many of them add up to a number, but I'm not good enough with this stuff to say anything about this.