Let $E \in \mathcal{L}$ and let $I_E$ denote the indicator function for $E$ in $\mathbb{R}^N$. Results

(i)	R2966: Indicator function under scaling of the argument
(ii)	R1242: Unsigned basic integral is compatible with measure
(iii)	R1166: Lebesgue measure under scaling

imply \begin{equation} \begin{split} \int_{\mathbb{R}^N} I_E (x / \lambda) \, \mu(d x) & = \int_{\mathbb{R}^N} I_{\lambda E} (x) \, \mu(d x) \\ & = \mu(\lambda E) = |\lambda|^N \mu(E) = |\lambda|^N \int_{\mathbb{R}^N} I_E (x) \, \mu(d x) \end{split} \end{equation} This establishes the first claim for measurable indicator functions $\mathbb{R}^N \to \{ 0, 1 \}$. The first claim for unsigned functions $\mathbb{R}^N \to [0, \infty]$ then follows by applying the principles in [[[x,125]]]. The second claim follows by applying the first claim to the constant $1 / \lambda$. $\square$