7.3. Measures

Example 7.3.3(e): Outer Measure of Intervals

Find the outer measure of the set A of all rational numbers in [0, 1]. Also show that for any finite collection of intervals covering A we have that the sum of their lengths is greater or equal to 1.
First, let's find the outer measure of the set A. The rational numbers in [0, 1] are countable so we can write the set A = { r1, r2, r3, ... }. For each rn define the set
Rn = (rn - 2-n/, rn + 2-n/)
Then the collection { Rn } is a countable cover of A with open intervals, so it is part of the infimum for computing m*(A). But
l(Rn) = 2-n =
Therefore m*(A) for every positive . But that means that m*(A) = 0.

Well, alright, the above proof is off by a factor 2 or so, but it does not matter if m*(A) or m*(A) 2 , so the prove is valid (fix the constants, though).

As for the second part, it is left as an exercise. Compare with some of the previous examples as a hint.

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