Casella & Berger (2001) write:

the Axiom of Countable Additivity, is not universally accepted among statisticians. ... [It] is rejected by a school of statisticians led by deFinetti (1972), who chooses to replace this axiom with the Axiom of Finite Additivity.

What might possibly be wrong or objectionable about the Axiom of Countable Additivity?

The Axiom of Countable Additivity is the 3rd condition below:

• "[T]here are finitely, but not countably, additive distributions that assign every natural number probability zero while still assigning the set of all natural numbers probability $1.$" Can you give an example or a link to one, please? Off the top of my head, I don't understand how this is possible. – saulspatz Nov 30 at 4:48
• Thanks very much. Rao and Rao looks too heavy for me; $40$ or $50$ years ago I'd have found it very interesting, I've no doubt. – saulspatz Nov 30 at 5:20