Every $L^1$ approximate identity converges to the identity in $L^p$ for $1\le p<\mathrm{\infty }$

Because $\int K_{\epsilon }=1$, one may write [L1, L3, given, algebra] $f\ast K_{\epsilon }-f={\int }_{{\mathbb{R}}^n}{K}_{\epsilon }(y)({\tau }_{y}f-f)dy\mathrm{.}$ Applying [L3] gives $\mathrm{\parallel }f\ast K_{\epsilon }-f{\mathrm{\parallel }}_p\le \int \mathrm{\mid }{K}_{\epsilon }(y)\mathrm{\mid }\mathrm{\parallel }{\tau }_{y}f-f{\mathrm{\parallel }}_{p}dy\mathrm{.}$

Let $\eta >0$ be arbitrary. If $\mathrm{\parallel }f{\mathrm{\parallel }}_p=0$, then $f=0$ in $L^p$ and step [L1, L2, L3, step 1.1, choose, algebra] 1.1 gives $\mathrm{\parallel }f\ast K_{\epsilon }-f{\mathrm{\parallel }}_p=0<\eta$ for every $\epsilon >0$. Assume now $\mathrm{\parallel }f{\mathrm{\parallel }}_p>0$, and put $M:={\sup }_{\epsilon }\mathrm{\parallel }{K}_{\epsilon }{\mathrm{\parallel }}_1<\mathrm{\infty }$. By [L2], choose $\delta >0$ so that $\mathrm{\parallel }{\tau }_{y}f-f{\mathrm{\parallel }}_p<\eta \mathrm{/}(2M)$ whenever $\mathrm{\mid }y\mathrm{\mid }<\delta$. By [L1], choose ${\epsilon }_0>0$ so that ${\int }_{\mathrm{\mid }y\mathrm{\mid }\ge \delta }\mathrm{\mid }{K}_{\epsilon }(y)\mathrm{\mid }dy<\frac{\eta }{4\mathrm{\parallel }f{\mathrm{\parallel }}_p}(0<\epsilon <{\epsilon }_0)\mathrm{.}$ For such $\epsilon$, split the integral from step 1.1 into $\mathrm{\mid }y\mathrm{\mid }<\delta$ and $\mathrm{\mid }y\mathrm{\mid }\ge \delta$. The near part is at most $\eta \mathrm{/}2$. For the far part, [L3] gives $\mathrm{\parallel }{\tau }_{y}f-f{\mathrm{\parallel }}_p\le 2\mathrm{\parallel }f{\mathrm{\parallel }}_p$, so ${\int }_{\mathrm{\mid }y\mathrm{\mid }\ge \delta }\mathrm{\mid }{K}_{\epsilon }(y)\mathrm{\mid }\mathrm{\parallel }{\tau }_{y}f-f{\mathrm{\parallel }}_{p}dy\le 2\mathrm{\parallel }f{\mathrm{\parallel }}_p{\int }_{\mathrm{\mid }y\mathrm{\mid }\ge \delta }\mathrm{\mid }{K}_{\epsilon }(y)\mathrm{\mid }dy<\eta \mathrm{/}2.$ Hence $\mathrm{\parallel }f\ast K_{\epsilon }-f{\mathrm{\parallel }}_p<\eta$ whenever $0<\epsilon <{\epsilon }_0$.

Because $\eta >0$ was arbitrary, $\mathrm{\parallel }f\ast K_{\epsilon }-f{\mathrm{\parallel }}_p\to 0$ as [step 2.1] $\epsilon \to 0^{+}$, proving the convergence in $L^p$.