I had a very simple question, that I cannot figure out:
Let $A \subseteq \mathbb{R}^d$ for some positive integer $d$. Suppose that $A$ has Lebesgue measure $0$. Let $Q$ be an orthogonal matrix. Then, does the set: $$Q A := \{Q a: a\in A\}$$ also have Lebesgue measure $0$? Further, is it true that multiplication by an orthogonal matrix preserves Lebsgue measure?
Any help will be appreciated.