Vtyjkyuk

Show that lemma 3.4 (b) in The Elements of Integration by Bartle may fail if the finiteness condition $mu(F_1)<+infty$ is dropped.

Lemma 3.4(b): Let $mu$ be a measure defined on a $sigma$-algebra $X$. If $(F_n)$ is decreasing sequence in $X$ and if $mu(F_1)<+infty$, then
$$mu Bigg(bigcaplimits_n=1^inftyF_nBigg)=limmu(F_n)$$

You can see that in the proof of Lema 3.4(b) when he defines the sequence of sets $E_n=F_1 - F_n $ , you can't say that $mu (E_n) = mu (F_1) - mu (F_n)$ if $mu(F_1)$ is not finite.

This could get you in trouble because expression like $infty - infty$ aren't well defined in this context.

For a counter-example, you could use $X=mathbbR$ as the set; the power set $mathcalP(mathbbR)$ as the $sigma$-álgebra and the following measure:

$$mu (E)=begincases vert E vert mbox, if $E$ is finite\
+inftymbox, otherwise
endcases$$

Then use the sequence $F_n=(-frac1n,frac1n)$.

$mu(bigcaplimits _n=1^inftyF_n)=mu (0)=1 $, but $limmu(F_n)=+infty$.

As Robson said, please include your thoughts on the question. You'll get better responses if we know what you are thinking already.

However, I thought the question was interesting, so:

Hint: Can any of the sets have finite measure if the lemma is to fail?