7.3. Measures

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))
b - a - 2/n = m*(Jn) m*(J) m*(closure(J)) = b - a
Hence b - a - 2/n m*(J) b - a for all n, which implies that m*(J) = b - a, as we suspected.
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