Let $\lambda^*$ and $\mathcal{L}(\Bbb{R})$ denote the Lebesgue measure and the Lebesgue sigma algebra on $\Bbb{R}$ respectively.
Let $\{A_k\}_{k \in \Bbb{N}} \subset \mathcal{L}(\Bbb{R})$ such that $[0,1] = \cup _{k\in \Bbb{N}}A_k$ and $\sum_{k\in \Bbb{N}}\lambda(A_k) < \infty$ . Then, the set $B$ of the elements which appears only finitely on $\{A_k\}_{k \in \Bbb{N}}$ satisfies $\lambda^*(B)=1$ .
In order to prove it, I am trying to understand $B$, but I am getting stuck. We can desbribe it as:
$$B=\{x \in [0,1] \, | \, \exists n_0 \forall n \geq n_0 : x \notin A_n \}=\{x \in [0,1] \, | \, \exists n_0 \forall n \geq n_0 : x \in A_n^c \}$$
Since, we can define the function $n_. : B \hookrightarrow \Bbb{N}, n_x = \min\{m \in \Bbb{N} \, | \, \forall n\geq m : x\in A_n^c\}$ which gives us an equivalence relation on $B$ : $x \sim y \leftrightarrow n_x=n_y$, so we have the partition $\{B_n\}_{n\in \Bbb{N}}$ of $B$ where $B_n=\{x \in B : n=n_x \}$
I suspect that $\lambda^*(B_n) \geq 1/n$ or something similar but I am very stuck and don't no how to continue or if it is better trying to prove that $\lambda^*(B^c)=0$ , so any help would be appreciated.
This is the Borel-Cantelli Lemma: https://en.wikipedia.org/wiki/Borel%E2%80%93Cantelli_lemma. I think the key step in making the proof truly "visible" is noticing that the set $B$ of elements that appear in infinitely many $A_k$ (in the language of probability theory, this is the [event]=[set of $x$] that infinitely many [events $A_k$ occur]=[sets $A_k$ contain $x$], i.e. the events $A_k$ occur infinitely often, which is often abbreviated "i.o." in probability contexts) is: $$[A_k \text{ i.o.}] = \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty A_k.$$ Another way to think about it is $$\textbf 1_{[A_k \text{ i.o.}]} = \limsup_{n\to\infty} \textbf1_{A_n}$$ (indeed $[A_k \text{ i.o.}]$ is often written $[\limsup A_k]$). See the Wikipedia page for the full proof if you need extra help after this hint.
There is an alternative slick proof via Fubini's theorem http://www.columbia.edu/~ks20/stochastic-I/stochastic-I-BC.pdf. That link has some interesting probability questions that can be solved using the Borel-Cantelli lemma. This pdf also has some https://ocw.mit.edu/courses/18-304-undergraduate-seminar-in-discrete-mathematics-spring-2015/2dc1c9e37d402c000b628ee85e2228d1_MIT18_304S15_project2.pdf.
The above links also discuss that this result has a partial converse: if we assume a further condition on the $A_k$ (a probabilistic notion called "independence of events"), the converse is true. That is known as the 2nd Borel-Cantelli lemma.