Examples 7.3.7(e): Measurable Sets
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Show that the interval
(a, )
is measurable.
Context
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:
-
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)
-
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.
-
(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.