Use image measures to give a new proof of Problem 5.9 , i.e. show that

\[\lambda^{n}(t \cdot B)=t^{n} \lambda^{n}(B) \quad \forall B \in \mathscr{B}\left(\mathbb{R}^{n}ight), \quad \forall t>0\]

Data from problem 5.9

Dilations. Mimic the proof of Theorem 5.8(i) and show that \(t \cdot B:=\{t b: b \in B\}\) is a Borel set for all \(B \in \mathscr{B}\left(\mathbb{R}^{n}ight)\) and \(t>0\). Moreover,