Interactive Real Analysis
- part of Examples 7.3.15(a): Monotone sequences of measurable sets
For decreasing sets we had to assume that m(A1)
was finite. Show that without this assumption the statement in the previous
proposition is false.
Context
We have to find a decreasing sequence of sets An such
that lim m(An) is not equal to
m(
Context
An).
Let An = [n, 
).
Then m(An) = for
all n, so in particular
lim m(An) = .
On the other hand we have that
An = 0
so that
m( An ) = 0.
That finishes the example.