My lecture notes on measure theory motivate countable additivity of measures by the method of exhaustion, which makes perfect sense to me.
After this the notes pose the question whether we could also demand uncountable additivity and then state the following theorem.
Theorem: Let $I$ be a set and $a: I \to (0,\infty)$ a map such that
$\sum \limits_{k \in I} a_k$ converges. Then $I$ is countable.
(Please note that the convergence of an uncountable sum can be defined in terms of nets (see here).)
The notes then mention that this means we only need countable additivity.
I understand the theorem and that this means that uncountable additivity of measures would mean that any uncountable union of measures would have infinity measure. But why is this a problem? Does it break anything? One thing I came up with as an explanation is that the measure of an uncountable sum does not really depend on the exact measures of the single sets. As long as uncountable many sets have positive measure the union will have infinite measure (unlike for a countable sum).
Thanks for any help and suggestions!
Edit: I know that in the end uncountable additivity would imply that every subset of $\mathbb{R}$ has Lebesgue measure $0$, but I feel like the author of the notes has something else in mind. In fact, at this stage we don't even know the Lebesgue measure.
Best Answer
Say $\mu$ is an uncountably additive measure on some set $X$ and $\mathcal{S}\subseteq \mathcal{P}(X)$ is an uncountable collection of measurable sets. Then since expressions such as: $$\mu\left({\bigcup \mathcal{S}}\right)=\sum_{A\in \mathcal{S}}\mu\left(A\right)$$ would now be meaningful, using a $\sigma$-algebra $\mathfrak{m}$ to represent the domain of $\mu$ is no longer sufficient, as we have no guarantee that ${\bigcup \mathcal{S}}\in\mathfrak{m}$.
If we try to extend $\mathfrak{m}$ so as to make equalities such as the above meaningful, we'd be unable to develop any meaningful measure that naturally extends the concepts of length/area/volume, since sets that cannot be measurable would be considered measurable with these definitions.