The Third Axiom of Probability has been a well-accepted axiom yet there seems to be a nice proof of it. I couldn't trace an implicit assumption of this axiom anywhere in the proof. I'll write a sketch of the proof; please tell me if it could be replaced by it.
Thm : Let $\{E_1, E_2, \cdots\}$ be mutually exclusive events of some countable set $\Omega$ such that $P(\Omega) = 1$. Then $P(E) = \displaystyle\sum_{i=1}^{\infty} P(E_i)$.
Proof : The case when $E$ is finite can be proven easily (from definition) and hence is left out. (Edit : Added as comment #1) When $E$ is countable, choose some $\epsilon > 0$. Denote $\displaystyle\sum_{i=1}^{\infty} P(E_i)$ by $\text{P}$ and $F_n = \displaystyle\sum_{i=1}^{n} P(E_i)$.
Observe that we can get some $n$ in $\mathbb{N}$ such that $|P – F_n| \leq \epsilon$ (which basically follows from the monotone convergence thm and def. of convergence). Additionally, we can yield some (finite set) $E^{*}_i \subseteq E_i$ for each $1 \leq i \leq n$ so that $P(E_i) \leq P(E^{*}_i) + \epsilon$.
Obviously $\displaystyle\bigcup_{i=1}^{n} E_i^{*} \subseteq E$ and so by Boole's inequality implies $P\left(\displaystyle\bigcup_{i=1}^{n} E_i^{*}\right) \leq P(E)$. From all these above, it follows that :
$$\text{P} \leq F_n + \epsilon \leq P\left(\displaystyle\bigcup_{i=1}^{n} E_i\right) + \epsilon \leq \displaystyle\bigcup_{i=1}^{n} P(E_i) + \epsilon \leq \bigcup_{i=1}^{n} P(E_i^*) + (n+1)\epsilon \leq P(E) + \epsilon'$$.
This shows that $\text{P} = \displaystyle\sum_{i=1}^{\infty} P(E_i) \leq P(E)$.
We can also show the reverse to eventually conclude the equality of $\text{P}$ and $P(E)$. (I'm not attaching the proof, unless asked in comments to keep the size of the post decent)
Why is it an axiom and not a theorem then?
I would like to add that I have no background in measure theory and hence is not defined on a measure theoretic space in particular.
EDIT (as suggested by @Bungo in comments) : Please note that the $\Omega$ is assumed to be countable. I'm trying to see if setting $P(A \cup B) = P(A) + P(B)$ as the axiom ($A \cap B = \phi; A,B \subseteq \Omega$) proves both the finite and the infinite case.
Best Answer
Countable additivity of a probability measure can be proven as a theorem if we assume what some authors call left continuity of measures as the third axiom instead: if $A_n \supset A_{n+1}$ is a decreasing sequence of events with $\cap_n A_n=\emptyset$ then $\lim_{n\to \infty} P(A_n)\to 0$.
In fact one can prove $P$ is left continuous if and only if $P$ is countably additive. For context, Kolmogorov took left continuity as an axiom and proved countable additivity when he axiomatized probability theory in 1933 (and if I recall correctly, showed the equivalence as well). Ignoring matters of pedagogy, efficiency of proofs, and applications, it is then merely a matter of taste which to pick as the axiom and which to derive as the theorem, since they are equivalent. It seems most modern texts choose countable additivity as the axiom, however.
All we must do to show this is impossible is find one probability measure on a countable sample space $\Omega$ that is finitely additive but not countably additive, and then we know that finite additivity cannot, in general, imply countable additivity. This is apparently not trivial to do even on $\Omega = \mathbb{N}$, as it requires, in some form, the axiom of choice or ultra-filters. See here, here and here for more details about the set-theoretic issues and the examples using ultra-filters, the natural density, and/or AC. In summary, the counterexamples we want exist but cannot be explicitly constructed.
Please comment for clarification or if I've made any mistakes!