Why the Lemma below is true even if the measure $\mu$ is infinite?

Question

A Bartle exercise proposes to show that lemma 8.8 is true even if the measure $\mu$ is infinite. What I can not see is any difference between the proof of lemma 8.8 and the solution for exercise. Can someone explain to me if it has any difference?

Answer 1

You're right: the given argument works without assuming $\mu$ is finite. The only place that such finiteness might be used would be to conclude that$$\mu\left(\bigcap F_n\right)=\lim\mu(F_n).$$ However, since we know $\mu(F_1)<1/2$ is finite, this equation is valid even without knowing that $\mu$ is a finite measure.

(On the other hand, the assumption that $\lambda$ is finite is crucial to the argument, since we do not know that $\lambda(F_n)$ is finite for any $n$. Indeed, the result is not true without some kind of finiteness assumption on $\lambda$. For instance, let $\mu$ be Lebesgue measure on $\mathbb{R}$ and let $\lambda(A)=0$ if $\mu(A)=0$ and $\lambda(A)=\infty$ if $\mu(A)>0$.)