The image $f(A)$ of a measurable subset $A \subset \mathbb{R}$ is also measurable under $f(x) = ax + b$

Question

Let $f:\mathbb{R} \longrightarrow \mathbb{R}$ be defined by $f(x) = ax + b$ with $a>0$. I want to show that if $A \subset \mathbb{R}$ is measurable, then $f(A)$ is also measurable.

Would anyone give me an idea to prove the statement? Thanks.

Answer 1

Hint: $f$ is a homeomorphism so it preserves Borel sets. The null sets are easy to handle too, as intervals are scaled with a factor $a$.