Proof of translation and linear transformation invariance of R_k

Question

Let A be a Lebesgue measurable set. Suppose that $f$ and $g_r$ are respectively the translation and linear tansformation with ratio $r$ on $\mathbb{R}_k$. Prove that $f(A)$ and $g_r(A)$ are Lebesgue measurable sets and

$m_k(f(A))=m_k(A)$

$m_k(g_r(A))=m_k(A).r$

$m_k(x+A)=m_k(A)$

$m_k(rA)=\vert r \vert .m_k(A)(r \neq 0)$

I start from scratch and what the idea of the begining of the proof ? Thank you