Let $M = (\mathbb{R}^N, \mathcal{L}, \ell)$ be a D1744: Lebesgue measure space such that

(i)	$\eta = \{ \eta_{\varepsilon} \}_{\varepsilon \in (0, \infty)}$ is a D138: Standard mollifier for $\mathbb{R}^N$

Let $\varepsilon > 0$. By definition \begin{equation} \eta_{\varepsilon}(x) = \varepsilon^{-N} \phi \left( \frac{x}{\varepsilon} \right) \end{equation} where $\phi(t) = e^{- \pi |t|^2}$. Consider an endomorphism $\varphi : \mathbb{R}^N \to \mathbb{R}^N$ given by $\varphi(x) = \varepsilon x$. This function is differentiable with $|\mathsf{det}(D \varphi)| = \varepsilon^N$. Instituting a change of variables in the sense of R2143: Differentiable change of variables for signed Lebesgue integral with respect to this endomorphism and applying results

(i)	R4619: Homogeneity of basic Riemann integral
(ii)	R2254: Riemann integral of standard real gaussian function

yields \begin{equation} \begin{split} \int_{\mathbb{R}^N} \eta_{\varepsilon}(x) \, \ell(d x) & = \varepsilon^{-N} \int_{\mathbb{R}^N} \phi \left( \frac{x}{\varepsilon} \right) \, \ell(d x) \\ & = \varepsilon^{-N} \int_{\mathbb{R}^N} \phi(x) |\mathsf{det}(D \varphi)| \, \ell(d x) \\ & = \varepsilon^{-N} \int_{\mathbb{R}^N} \phi(x) \varepsilon^N \, \ell(d x) \\ & = \varepsilon^{-N} \varepsilon^N \int_{\mathbb{R}^N} \phi(x) \, \ell(d x) \\ & = \int_{\mathbb{R}^N} \phi(x) \, \ell(d x) \\ & = 1 \end{split} \end{equation} $\square$