According to S. Bochner, finitely additive measures are more interesting,
and perhaps more important, than countably additive ones (see Maharam
(1976)). Finitely additive measures arise quite naturally in many areas of
analysis. Over the years, there has been a sustained growth of activity in
finitely additive measures propelled by mathematicians and statisticians.
The case for finitely additive probability is put forward strongly by Dubins
and Savage (1965) in their book “How to Gamble If You Must”. They
refer to de Finetti, who, in a large number of papers published as early as
1930, “has always insisted that countable additivity is not an integral part
of the probability concept but is rather in the nature of a regularity
hypothesis.” In fact, Dubins and Savage “view countably additive measures
much as one views analytic functions-as a particularly important special
case.” But not much attention is paid to finitely additive measures in
text-books on Measure Theory. (Books on Functional Analysis do a bit
better.) One reason could be that countably additive measures are more
tractable than finitely additive ones.