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.