An approximate identity is a function $\phi_{\epsilon} \in L^1(\mathbb{R}^n)$ with the following properties:
(a) $\| \phi_{\epsilon} \|_{L^1(\mathbb{R}^n)} \leq c$ for some constant $c>0$.
(b) $\int_{\mathbb{R}^n} \phi_{\epsilon} dx =1$.
(c) for $\delta >0$ it holds that $\int_{|x| \geq \delta} |\phi_{\epsilon} (x) | dx \rightarrow 0$ as $\epsilon \rightarrow 0$.
I would like to find an approximate identity for any function $\phi$ with the property $\int_{\mathbb{R}} \phi(x) dx=1$. I therefore defined
$\phi_{\epsilon}(x)=\frac{1}{\epsilon} \phi(\frac{x}{\epsilon}).$
I would now like to show the three properties stated above for my newly defined function $\phi_{\epsilon}$. I’m only having troubles with (a).
For (a), I just computed the integral and with substitution I derived that $\|\phi_{\epsilon}\|_{L^1(\mathbb{R})} =\|\phi\|_{L^1(\mathbb{R})}$.Is that enough? I know that $\int_{\mathbb{R}} \phi(x) dx=1$ by assumption. But does this also imply that $\phi \in L^1(\mathbb{R})$?
Both (b) and (c) also follow easily by just computing the integrals and using the substitution $y=x/\epsilon$.