I got stuck on a problem on Folland's book, here is the problem:

suppose $\displaystyle \mu$ is a Radon measure on $\displaystyle X$ such that $\displaystyle \mu(\{x\})=0$ for all $\displaystyle x\in X$, and $\displaystyle A$ is a Borel set in X such that $\displaystyle 0<\mu(A)<\infty$. Then show that any $\displaystyle \alpha$ such that $\displaystyle 0<\alpha<\mu(A)$, there is a Borel set $\displaystyle B \subset A$ such that $\displaystyle \mu(B)=\alpha$.

thank you all!