Need for a set to have finite Lebesgue measure.

Question

I'm reading Royden and Fitzpatrick's book (4th edition) wherein on page 44, we have the following theorem:

If $\{B_k\}_{k=1}^\infty$ is a descending collection of Lebesgue measurable sets and $m(B_1)<\infty$ then $m(\bigcap_{k =1}^\infty B_{k})=\lim_{k\rightarrow\infty}m(B_k)$.

At the end of this section, there is an exercise: Show at the assumption that $m(B_1)<\infty$ is necessary. I'm trying to find a counter example where if $m(B_1)$ is not finite, then the above equality fails to holds.

Can anyone gice any ideas as to how one would construct such a collection?

Answer 1

Counterexample.

Let $B_n:=[n,\infty)\subseteq\mathbb R$ for $n=1,2,\dots$

Let $m$ be the Lebesgue measure on $\mathbb R$.

Then $(B_n)_n$ is a descending sequence of Lebesgue measurable sets, and $B:=\bigcap_{n=1}^{\infty}B_n=\varnothing$.

So: $$m(B)=m(\varnothing)=0\neq+\infty=\lim_{n\to\infty}m(B_n)$$

Answer 2

Try $B_k = [k,\infty)$ with Lebesgue measure on the line.