## Using Measurable Covers and Kernels II

Following yesterday’s post, here are some more useful facts that we can prove with the help of measurable covers and measurable kernels.

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.