Using Measurable Covers and Kernels II
If and are disjoint sets in , then
Here, we take to be a measurable cover of and to be a measurable kernel of . The difference must be contained in , and so
On the other hand, we can take to be a measurable kernel of and to be a measurable cover of . Now the difference is contained in , and we find
Now, if , then for every whatsoever, we have
We can take and and stick them into the previous result to find
But since , we know that , and this establishes our result.
Interestingly, we can use this method of inner measures as an alternative approach to our extension theorems. If is a -finite measure on a ring , and if is the induced outer measure on , then for every set of finite measure and every we have
Then if and are two sets in such that , then we find
and so we can use this formula as the definition of the inner measure . Then we can define a set with to be -measurable if the inner and outer measures match: . And from here, the rest of the theory is as before.