Example 7.3.3(c): Outer Measure of Intervals
Find the outer measure of an open interval (a, b)As before, we expect that m*(a, b) = b - a.
Set J = (a, b) and define Jn = [a+1/n, b-1/n] for n a positive integer. Since Jn is closed, we know by the previous example that m*(Jn) = b - a - 2/n. Also
Jn J closure(J)so that by an earlier example
m*(Jn) m*(J) m*(closure(J))But
b - a - 2/n = m*(Jn) m*(J) m*(closure(J)) = b - aHence b - a - 2/n m*(J) b - a for all n, which implies that m*(J) = b - a, as we suspected.