The book I'm reading (Mathematical Foundations of Statistical Mechanics by Khinchin) refers to a "known theorem of the metric theory of sets" (page 21 of the first edition if anyone is wondering).
It seems to me by how the author uses it, that the theorem is:
Let $V$ be a subset of $\mathbb{R}^n$ with finite Lebesgue measure, and let $A_n$ be a sequence of measurable subsets of $V$ such that $\sum_{n=1}^\infty \lambda(A_n) < \infty$, then the set of points of $V$ that belongs to infinitely many $A_n$ has measure $0$.
By now I only managed to prove that given a subsequence $A_{\sigma_n}$ of $A_n$, it must be $\lambda(\bigcap_{n=1}^\infty A_{\sigma_n}) = 0$, since if $\lambda(\bigcap_{n=1}^\infty A_{\sigma_n}) = \delta > 0$, then $\sum_{n=1}^\infty \lambda(A_n) \geq \sum_{n=1}^\infty \lambda(A_{\sigma_n}) \geq \sum_{n=1}^\infty \lambda(\bigcap_{i=1}^\infty A_{\sigma_i}) = \infty$.
However here I'm stuck, since the cardinality of stricly increasing sequences of natural numbers is $\mathfrak{c}$. I dont even know if this is a theorem at all to be honest, let me know if someone knows a proof of this, and if this is.
If however this is not a theorem I will add the entire line of reasoning present in the book, to figure out which theorem is the author referring to.
EDIT:
If anyone is wondering, as AnneBauval says in the comment section, this is infact a theorem, and it is known as Borel-Cantelli lemma: https://en.wikipedia.org/wiki/Borel%E2%80%93Cantelli_lemma .
The key to apply this lemma is to notice that the set I was trying to measure is infact the limsup of the sequence $A_n$.
