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.