Does it make sense to define the length of a line segment in terms of addition of infinite points?

Question

Since a point has zero length, how can a line segment of, say, 1-unit length—which is a collection (addition) of infinite points, that is $0 + 0 + \cdots$—have 1-unit length? Does it make sense to say $0 + 0 + \cdots = 0$?

Answer 1

If a set is the union of countably many disjoint intervals, the total length of the set is the sum of the lengths of the intervals. However, it is not so for uncountably many, precisely because of your example. This corresponds to the fact that Lebesgue measure is countably additive.