Problem for measure theory

Question

Suppose that $A_i$ is a measurable set in $\Bbb R^n$ for each $i$ belonging to an index set $I$, and suppose that the $A_i$'s are disjoint and that $m(A_i)\gt 0$ for all $i$. prove that $I$ is countable

MY Attempt : Since All $A_i$ is disjoint and subset of $\Bbb R^n$ such that there is rational number for each $A_i$. So rational number is countable such $A_i$ is countable. but I can't satisfy second condition $m(A_i)\gt 0$. any help please

Answer 1

$\newcommand{\pr}[1]{\left(#1\right) }$

Hint: let $I_{m,n}$ be the set of all indexes $i$ such that $m\pr{A_i\cap B\pr{0,m} }\gt 1/n$, where $B\pr{0,m}$ denotes the ball of center $0$ and radius $m$. Then you have to show that:

1. each $I_{m,n}$ is finite and
2. $\bigcup_{m,n\in\mathbb N}I_{m,n}=I$.