Interactive Real Analysis
- part of 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.
Context
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
ContextRn = (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, rn + 2-n/
Therefore m*(A)l(Rn) =
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.