The context in which I ask this question is de Finetti's subjective probability theory, where he takes that personal probabilities should be finitely additive and there is no need to impose the stronger condition of countable additivity. I was trying to understand how much set theory he needs at the foundational level for his conviction.

Thanks, this is very nice. The next question is this. WITHOUT AC, whether or not it can be shown that there does not exists any nontrivial FINITELY additive measure defined over all subsets of the reals? Is there a result showing that ZF + {nonexistence of finitely additive measure over all subsets of reals} is consistent?

Yes, finitely additive probability measure.