There are a bunch of useful facts that we can prove with the help of measurable covers and measurable kernels. These will allow us to move statements we want to show about outer and inner measures to the realm of proper measures, where we can use nice things like additivity.
For example, given a pairwise disjoint sequence , we can show that
Just pick a measurable kernel for each . Then we can use countable additivity to show
For another, given a set and a disjoint sequence whose union is , then
This time, let be a measurable kernel of , so that
On the other hand, is the union of the , and so we can use the previous result to get the opposite inequality.
Next: if , then clearly . But conversely, if , then . To see this, let be a measurable kernel and be a measurable cover of . Then we calculate
But , so (by the completeness of ), and thus . Thus sets of finite measure in are exactly those in for which the outer and inner measures coincide.
Interestingly, notice how the last step of this proof echoes our earlier result that a set is Jordan measurable if and only if the Jordan content of its boundary is zero.