Question

prove

that m∗ (A ∪ B) ≤ m∗ (A) + m∗ (B) for any

A, B ⊂ R. For this problem, use the definition of m∗ to formally prove that this is true?

Answer 1

Let { A,B} =Fi be a finite collection of pairwise disjoint bounded closed sets .and let C=.

Then m(C)= = m(A) +m(B)

Now let us consider open sets Gi ⊇ Fi= {A,B} such that m(Fi) > m(Gi) − ε /2 , for i ∈ {1, 2}.

Then we have  m(A) + m(B) > m(G1) + m(G2) − ε.

From this inequality we obtain

m(A ∪ B) ≤ m(G1 ∪ G2) ≤ m(G1) + m(G2) < m(A) + m(B) + ε.

[ as If a bounded open set G is the union of a finite or countable family of open sets {Gi}i∈J , then

and Let U and V be bounded sets of real numbers such that each of these two sets is either open or closed. Then U ⊆ V implies m(U) ≤ m(V ). Applying these two theorem we got the above inequality.]

Hence, m(C) ≤ m(A) + m(B).