Let \(\phi\in L^1(\RR^n)\) with \(\int_{\RR^n}\phi = 1\), and for \(t>0\) let \(\phi_t(x)\coloneqq t^{-n}\phi(t^{-1}x)\) be its dilations. Then \(\phi\), or the family \((\phi_t)_{t>0}\), is called an approximate identity.
Let \(\phi\in L^1(\RR^n)\) with \(\int\phi=1\). Then:
\(\int \phi_t = \int\phi = 1\) and \(\norm{\phi_t}_1 = \norm{\phi}_1\) for every \(t>0\); in particular \(\norm{\phi_t}_1=1\) when \(\phi\geq 0\).
\(\sup_{t>0} \norm{\phi_t}_1 < \infty\).
For every \(h>0\), \(\lim_{t\to 0^+} \int_{\abs{x} \geq h} \abs{\phi_t(x)}\dx = 0\).
The substitution \(x=ty\), \(dx=t^n\,dy\), gives \(\int\phi_t=\int\phi\) and \(\norm{\phi_t}_1=\norm{\phi}_1\), which proves the first two items. The same substitution gives \[ \int_{\abs{x}\geq h}\abs{\phi_t(x)}\dx=\int_{\abs{y}\geq h/t}\abs{\phi(y)}\,dy, \] and the right side tends to \(0\) as \(t\to 0^+\) by dominated convergence, with dominating function \(\abs{\phi}\in L^1\).