Property of volume

Question

Let $\mu$ be the Lebesgue measure on $\mathbb{R}^n$. I have to say that I'm not an expert in measure theory, I wonder if is true that if $\mu(A)\leq \mu(B)$ then for every $C,C'$ such that $\mu(C)\leq \mu(C')$, then $\mu(A\cap C) \leq \mu(B\cap C')$?

Thanks in advance.

Answer 1

Νο it is not true.

Take for instance $A=[0,1]$,$B=[0,10]$, $C=[0,1]$, $C'=[10,20]$