I'd strongly suggest moving this to the Math.SE forum, see this post, but seeing as how you are keeping it here, I'll offer some thoughts:
Oppy is discussing what properties a measure should possess, from a foundational perspective. Note that a measure is defined on a measurable space, which consists of a set, a sigma-field on that set, and a set-function that obeys the measure axioms:
- The measure of the empty set is 0
- If set A is a subset of set B, the measure of set B should be at least as large as the measure of set A
- The measure of the (finite or countable) union of mutually disjoint sets is the sum (finite or countable) of the measure of each such set.
Note that (3) is an assumption, and one that Oppy justifies by appeal to a very basic mathematical property that we take for granted...the limit of infinite sums. I have seen other authors (e.g., Robert Ash) also take pains to justify the need for countable additivity by appealing to the richness of the mathematics allowed by assuming/requiring countable additivity, not on any physical grounds.
In fact, there are set functions that are finitely additive (all partial sums exist) but that don't converge to the value of the set function at the limit itself (think jump discontinuity) when a countable sum is taken.
Specifically for your question, Oppy was addressing Zeno's paradox about distance. The distance to be traveled is being divided into an infinite sequence of intervals, each half as long as the previous one. Now, under the axioms of measure, we'd expect that no matter how we divide up the distance to be covered, as long as we specify a countable set of divisions, the sum of the length of the divisions should give the total length (i.e., a meter is a meter, no matter what scale we measure it in or how we break it up).
Thus, if we reject countable additivity, then we can have measures (e.g., length) with counterintuitive properties, such as the total distance covered by the runner being dependent on the way we break up and add the partial distances. By requiring countable additivity, we are forcing a common-sense definition of length (and measure in general) and excluding definitions that lead to weird behavior.
There are several interesting counterexamples that show what can happen if we try to work with non-measurable sets (Banach-Tarsky paradox, Vitali Sets) or if you try to work with set functions that are only finitely additive