The following problem is from Papa Rudin, exercise 2.15; I'm trying to rephrase it so that you won't have to look it up, but there's a chance that the rephrasing will be wrong since I'm not very good at this stuff. I just need a check on this, but I'm not very confident in my answer at all.
Problem: Let be a regular measure; let E be measurable, . Show that for disjoint compact sets and .
Attempted Solution Sketch: Choose by using the (inner) regularity of (and finite measure of E):
where in all cases .
Then, and the are disjoint by construction. Thus, . Fix and find such that . Then
Because was arbitrary, , which implies equality. Take , and the result follows.
Postscript: The details are, possibly, fudged a little bit, but I'm okay with that as long as the main idea is there. I'm not sure this works, as I'm not as comfortable with some of the material here. I'm also not sure if there is a more direct route to this.