If $A \in \mathbb{R}$ with $\lambda(A)=0$ then A has finitely many elements

Question

I am trying to determine if $A \in \mathbb{R}$ with $\lambda(A)=0$ then A has finitely many elements

My intuition is that this is FALSE. Because if $A=\mathbb{Q}$ then $$\lambda(A)=\lambda(Q_1)+\lambda(Q_2)+...+\lambda(Q_n)$$ for $n \rightarrow \infty$

However on the other hand, then $\mathbb{Q}$ is countable so I am not sure if that affects that it is actually finite

Any hint would be appreciated

Answer 1

Your example is fine. You have found an infinite set with measure $0$. $\mathbb N$ is another counter-example.