## Examples 7.3.15(a): Monotone sequences of measurable sets

For decreasing sets we had to assume that

We have to find a decreasing sequence of sets *m(*was finite. Show that without this assumption the statement in the previous proposition is false.**A**_{1})*such that*

**A**_{n}*lim m(*is not equal to

**A**_{n})*m(*.

**A**_{n})
Let *A_{n} = [n, )*.
Then

*m(*for all

**A**_{n}) =*n*, so in particular

*lim m(*.

**A**_{n}) =
On the other hand we have that
* A_{n} = 0*
so that

*m(*. That finishes the example.

**A**_{n}) = 0