solvet please Let the sets A and B have volume Then show A u

solvet please:

Let the sets A and B have volume. Then show A union B has volume. If A intersection B and.A\\B have volume as well, then prove v(A union B) = v(A) + v(B) - v(A intersection B); v(A\\B) = v(A) - v(B) if A subset B.

Solution

A and B have volumes implies there exists anincreasing sequence of measurable sets A[n], B[n] , all with finite measure tending to A and B in measure,Using the sequence A[n]UB[n] , we conclude that AUB has finite volume as well.

(a) and (b) follow from the basic axioms fo measures of disjoint unions.(use Venn diagrams)