 Countable additivity

For pairwise disjoint measurable sets $E_n$, countable additivity means

The infinite series is the limit of its nonnegative partial sums. This controls an entire countable union, not merely each finite stage. If $q_1,q_2,\dots$ enumerate the rational numbers in $[0,1]$, their finite collections have Lebesgue measure zero, and countable additivity gives their whole union Lebesgue measure zero. Their being a dense set does not change this calculation.

The axiom makes no assertion about an arbitrary uncountable set of disjoint pieces. In particular, $[0,1]$ is the union of its singletons, each of Lebesgue measure zero, yet its Lebesgue measure is one. Extending additivity to all such unions would contradict interval length.