MathCS.org - Real Analysis: Example 7.3.3(c): Outer Measure of Intervals

Example 7.3.3(c): Outer Measure of Intervals

Find the outer measure of an open interval (a, b)

Context

As before, we expect that m^*(a, b) = b - a.

Set J = (a, b) and define J_n = [a+1/n, b-1/n] for n a positive integer. Since J_n is closed, we know by the previous example that m^*(J_n) = b - a - 2/n. Also

J_n J closure(J)

so that by an earlier example

m^*(J_n) m^*(J) m^*(closure(J))

But

b - a - 2/n = m^*(J_n) 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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Example 7.3.3(c): Outer Measure of Intervals