Mollifier and Approximation of Identity

Let $\eta\in C_c^\infty(\mathbb R^n)$ with a support contained in the unit ball, be a probability density function, that is, $\eta\geq 0$ and $\int_{\mathbb R^n}\eta(x)\mathrm dx=1$. We call $\eta$ a mollifier.

For any $\epsilon>0$, we define the function $\eta_\epsilon:\mathbb R^n\to\mathbb R$ by

$$\eta_\epsilon(x)=\frac 1{\epsilon^n}\eta\left(\frac x\epsilon\right),\quad x\in\mathbb R^n.$$

Lemma. If $f\in L^1(\mathbb R^n)$, then $\|f*\eta_\epsilon-f\|_{L^1(\mathbb R^n)}\to 0$ as $\epsilon\to 0$.

Theorem. Weak.