Interactive Real Analysis - part of

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Examples 7.3.7(e): Measurable Sets

Show that the interval (a, ) is measurable.
We need to show that for every set A we have that
m*(A) m*(A (a, )) + m*(A (-, a])
because comp(a, ) = (-, a]. If m*(A) is infinite, there's nothing to prove. Therefore we can assume that m*(A) is finite. Then, because of the definition of outer measure as an infimum, there exists a countable collection of open intervals In that cover A and
l(In) m*(A) +
for any > 0. Define sets Jn and Kn as
Jn = In (a, )
Kn = In (-, a])
Then we have the following properties:
  1. Jn and Kn are intervals (or empty) so that
    m*(Jn) = l(Jn) and m*(Kn) = l(Kn) and
    l(Jn) + l(Kn) = l(In)
  2. Jn In and Kn In so that
    l(Jn) l(In) and l(Kn) l(In)
    In particular, all sums are absolutely convergent because the measure of A is finite.
  3. (A (a, )) Jn and (A (-, a]) Kn so that
    m*(A (a, )) m*( Jn) l(Jn) and
    m*(A (-, a]) m*( Kn) l(Kn)
    because of subadditivity and (1).
But then we have that
m*(A (a, )) + m*(A (-, a]) l(Jn) + l(Kn) =
      l(Jn) + l(Kn) = l(In) m*(A) +
Since this inequality holds for every > 0, it implies what we wanted to prove.
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