Translation invariance of the Lebesgue measure

How to prove that the Lebesgue measure $m$ is translation-invariant for all Lebesgue measurable sets?

I know that $m(B + x)=m(B)$, for Borel sets $B$ and $x \in \mathbb{R}^n$. So, I am trying to use the fact that for a Lebesgue measurable set $A$, there is a Borel set $B'$ such that $A \subset B'$ and $m(A)=m(B')$.

Here is my attempt: Write $A = B \cup N$ where $B$ is borel and $C$ is null. Then $A +x= (B+x) \cup (N +x)$ so that $$m(A)=m(B)=m(B+x) \leq m(A+x) \leq m(B +x) + m(N+x)$$

This shows that $m(A) \leq m(A+x)$. How do I show the other inequality?

Also, is it possible to prove that $A + x$ is Lebesgue measurable using only the Caratheodory condition. Any tips?