What happens when you drop countable additivity of a measure for countable additivity on compact sets?
Uniform probability distributions over the real numbers can't be defined within standard measure theory because measures need to be countably additive.
Dropping countable additivity for just finite additivity, you lose a lot of nice properties. Among others, I've heard that integrals end up reducing to just Riemann integrals again.
A different modification you could consider is dropping countable additivity for finite additivity, but maintaining countable additivity whenever all the sets being unioned are contained within some compact set. This should still allow you to define uniform probability measures, but it has more structure than just finite additivity.
Does anyone know of any research or discussions on this topic? What happens to integrals in this context? Presumably integrals over compact sets would be equivalent to regular Lebesgue integrals, but how about over the full space? Do integrable functions still form some nice Banach space?
Does anyone see any obvious issues with this kind of structure, or know of similar structures?