I understand the definition of a sigma ring (the infinite unions of elements in ring is still in said ring)
And I believe I also understand the definition of countably additive. For elements in ring with the infinite union of elements also in said ring. Then the function satisfy ..ect. Then the function is countably additive.
What I’m confused about is doesn’t this require measures to only exist on sigma rings? Since rings in general don’t have closure for infinite unions. But I know this isn’t true, the ring (collection of all finite unions of disjoint intervals) is a ring and not a sigma ring but it has a lebesgue measure.
I don’t know what I’m not understanding correctly.
Best Answer
You are mixing two things. For a measure to be countable additive you require that IF the countable union of disjoint measurable sets happens to be measurable then the measure of the union is the sum of the measures. For a sigma ring it is always the case that countable unions belong to the ring so you can dispense with the 'if'. But a general measure can be defined and even sigma additive over a more general substrate.