Let $\mathbb{R}^d$ be a D816: Euclidean real Cartesian product.
Let $\mu^*$ be the D780: Lebesgue outer measure on $\mathbb{R}^d$.
Let $\lambda \in \mathbb{R} \setminus \{ 0 \}$.

Then \begin{equation} \forall \, E \subseteq \mathbb{R}^d : \mu^*(\lambda E) = |\lambda|^d \mu^*(E) \end{equation}