Measures: Sigma-Additivity

Question

Disclaimer: Though this thread is written in a Q&A style any new thoughts are really welcome!

What reasons are there to restrict measures to countable additivity rather than uncountable additivity? Is there a deeper reason than just it "works"? Any scratchy ideas are welcome...

Answer 1

Mainly, sets and numbers rely on different footings.
That brings structural similarities but also structural discrepancies.
These allow one to study homomorphism as are measures only to a certain extend.

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To start with, both give rise to an algebraic structure by a binary operation admitting a neutral element: $$A\cup\varnothing=A=\varnothing\cup A$$ $$x+0=x=0+x$$

Now, these extend to finite operations due to associativity: $$A_1\cup\ldots\cup A_N$$ $$x_1+\ldots+x_N$$ (Also noting that commutativity holds for both!)
At this point finite additivity can be studied...

Next, the union immediately generalizes after a reformulation: $$A_1\cup\ldots\cup A_N=\{x\in X:x\in A_1\vee\ldots\vee x\in A_N\}=\{x\in X:\exists i\in\{1,\ldots,N\}:x\in A_i\}$$ arriving at the union over arbitrary collections: $$\bigcup_{i\in I}A_i:=\{x\in X:\exists i\in I:x\in A_i\}$$ The sum however requires the introduction of a Hausdorff topology.
After this first lack arriving at the sum over in principal arbitrary collections as well: $$\sum_{i\in I}x_i:=\lim_{J\subset I:\#J<\infty}\sum_{j\in J}x_{j}\quad J\leq J':\Leftrightarrow J\subseteq J'$$ (Note that both extensions are independent of reordering by construction!)

The main two differences however are that not every arbitrary collection of numbers has a convergent sum and if so then only countably many numbers can be nontrivial: $$\sum_{i\in I}x_i\text{ exists}\implies x_i\neq 0\text{ countably many}$$ The first point forces positive measures to lie within the extended positive real axis and complex measures within a bounded disk: $$\mu(A)\in\overline{\mathbb{R}}_+\text{ resp. }\mu(A)\in\mathbb{D}\subseteq\mathbb{C}$$ The second point is that tells us* (but doesn't force us**) to better consider merely countable collections.

*A bad situation usually mentioned: $\sum_{x\in[0,1]}\lambda(\{x\})=0\neq 1=\lambda(\bigcup_{x\in[0,1]}\{x\})$
**In fact, the Dirac measure possesses the uncountable additivity!

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So sigma additivity will be the most prominent candidate for structure preserving homomorphisms a.k.a. measures.

Answer 2

Uncountable additivity are considered especially in set theory.

For example, an uncountable cardinal $\kappa$ is a real-valued measurable cardinal if and only if there exists a nontrivial $\kappa$-additive probability measure on $\kappa$.

Although in some cases $\sigma$-additivity is enough. For example, the least cardinal which has a $\sigma$-additive probability measure is alway a real-valued measurable cardinal.

However, it should be noted that the existence of such a cardinal can not be proved in the usual foundation axiom of mathematics, $\mathsf{ZFC}$. Moreover, the statement that such a cardinal exists has very high consistency strength.