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.We have to find a decreasing sequence of sets An such that lim m(An) is not equal to m( 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.