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, )Then we have the following properties:
Kn = In (-, a])
But then we have that
- Jn and Kn are intervals (or empty) so thatm*(Jn) = l(Jn) and m*(Kn) = l(Kn) and
l(Jn) + l(Kn) = l(In)
- Jn In and Kn In so thatl(Jn) l(In) and l(Kn) l(In)In particular, all sums are absolutely convergent because the measure of A is finite.
- (A (a, )) Jn and (A (-, a]) Kn so thatm*(A (a, )) m*( Jn) l(Jn) andbecause of subadditivity and (1).
m*(A (-, a]) m*( Kn) l(Kn)
m*(A (a, )) + m*(A (-, a]) l(Jn) + l(Kn) =Since this inequality holds for every > 0, it implies what we wanted to prove.
l(Jn) + l(Kn) = l(In) m*(A) +