Show that $\mu(A\setminus B)=0$

Question

If $B\subset{A}$ and $\mu(A)=\mu(B)$. Prove that $\mu(A\setminus B)=0.$

If we have $\mu(A)<\infty$ then $\mu(A\setminus B)=\mu(A)-\mu(B)=0$.

But $\mu(A)=\infty$, I can't prove $\mu(A\setminus B)=0$.

Answer 1

It's not true if $\mu(A)=\infty$. For example, $A=[0,+\infty)$ and $B=[1,+\infty)$ gives $\mu(A\setminus B)=1$.

Or $A=\mathbb R$ and $B=[0,+\infty)$ gives $\mu(A\setminus B)=\infty$.