Prove that the following set is countable

Question

I am trying to solve the following question , I have a certain idea of the question but I am unable to formalise it effectively and take it all the way , please help

Suppose that $A_i$ is a measurable set in $R^n$ for each $i\in I$ where I is an index set. Also suppose that the $A_i$'s are disjoint and that $\lambda(A_i) > 0$ for all i ( $\lambda(A_i)$ stands for the lebesgue measure of A_i). Prove that the index set I is countable

My approach - So far I have tried to assume that all the $A_i$'s are contained in some fixed ball B(0,k) . Next since all the $A_i$'s have a non zero measure , this implies that only a finite number of them can be contained in the ball ( which has a finite measure )

Answer 1

The sets with $\lambda(A_i)>\frac{1}{n}$ form an at most countable collection for every positive integer $n$. So you're done.