We denote by $L^p (\Omega) (1 \leq p < \infty)$ the space $L^p (X, \mu)$, where $(X, \mu)$ is the Lebesgue measure space corresponding to an open set $\Omega$ in $R^n$. Let $u \in L^p (\Omega)$, and define
$$(J_\epsilon u)(x) = \epsilon^{-n} \int_\Omega \rho \left( \frac{x-y}{\epsilon} \right) u(y) dy$$ With $\epsilon > 0$.
We call $J_\epsilon u$ a mollifier of u. Prove that
(i) $J_\epsilon u$ is in $C^\infty (R^n)$
(ii) If u vanishes outside a subset $A$ of $\Omega$, then $J_\epsilon u$ vanishes outside an $\epsilon-neighborhood$ of $A$ [that is, outside the set $\{ x ; \rho (x,A) < \epsilon\} ]$
(iii) If $K$ is a closed subset of $\Omega$ with $\rho (K, R^n - \Omega ) \geq \delta > 0$, then
$$(J_\epsilon u)(x) = \epsilon^{-n} \int_{|y-x|<\epsilon} \rho \left( \frac{x-y}{\epsilon} \right) u(y) dy = \int_{|z|<1} \rho (z) u(x- \epsilon z)dz$$
for any $x \in K$, provided $\epsilon < \delta$
It is an exercise of the book "Foundation of modern analysis, Avner Friedman" exercise (3.3.2). Could you give me tips on how to prove this? I still can't get anywhere. Thanks you in advance