Let \(\phi\in L^1(\RR^n)\) with \(\int_{\RR^n} \phi = 1\), and for \(t>0\) let \(\phi_t\) be its dilation \(\phi_t(x)=t^{-n}\phi(t^{-1}x)\). Then \(\phi\), or the family \((\phi_t)_{t>0}\), is called an approximate identity.
For every \(\phi\in L^1(\RR^n)\) and \(t>0\), the substitution \(x=ty\) gives \(\int \phi_{t} = \int \phi\), so every dilation of an approximate identity also has integral \(1\).