In order to define a (countably additive) probability, we need a $\sigma$-algebra. I'm interested in finitely (but not countably) additive probability measures. My questions are:
(1) Do we need a $\sigma$-algebra to define a finitely additive probability measure or will an algebra suffice?
(2) What does integration look like with respect to a finitely additive probability measure? More specifically, does it make any sense to have a finitely additive probability measure on an uncountable set and integrate with respect to that measure? I don't want to actually integrate anything, I'd just like to know what makes sense/what is and isn't possible.
I don't have much of a background in measure theory so I'm not sure if these questions make much sense or how difficult they are to answer. Any thoughts would be greatly appreciated!
Best Answer
You don't need a sigma algebra for defining finitely additive measures. An algebra will do. There is quite a bit of work on finitely additive measures, especially finitely additive probability measures. We can define integrals of bounded measurable functions w.r.t. finitely additive measures using that fact that they are uniform limits of simple functions.