I have an exercise in my measure theory class which goes as following:
If $(X,A,\mu)$, where $A$ is a $\sigma$-algebra, is a measure space and $(B_n),n=1,2,\ldots$ a sequence of sets in $A$. Show that if $\mu(B_n)\to 0$ and $\sum\limits_{n=1}^{\infty}\mu(B_{n+1} - B_n) < +\infty$, then $\mu(\limsup B_n)=0$.
(Where $\limsup B_n:=\bigcap_{N=1}^{\infty}\bigcup_{n=N}^{\infty} B_n$.)
I’ve had some thoughts but I can’t gather them to end in a solution.
For example, I know from the Borel-Cantelli lemma that $\mu(\limsup (B_{n+1} - B_n)=0$, but I just can’t connect the dots for this one. Every help would be gladly appreciated. I’m very interested in the solution of course, but I’m more interested in understanding the thought behind it…