I want to show that if $A \in \mathcal{B}(\mathbb{R})$ and $\lambda(A)>0$ then A has infite elements
If $A=\emptyset$ or $A=\{a\}$ then it is clear that $\lambda(\{a\})=a-a=0$
But for $A=(a,b)$ for b>a I am not sure how to show this. Of course $(a,b) \in \mathcal{B}(\mathbb{R})$ and we know that $\mathbb{R}$ is uncoutable.
But is there some easier way to show this? I want to avoid a big (Cantor) proof to show that (a,b) is uncountable