I’m reading Evan’s PDE. And I got stuck in the proof of properties of mollifier(Pg. 715). The property is that:
(iv) If $1\leq p<\infty$ and $f\in L^p_\text{loc}(U)$, then $f^\varepsilon \to f$ in $L^p_\text{loc}(U)$.
at the step 4 attempting to control $\lVert f^\varepsilon\rVert_{L^p(V)}$ by $\lVert f\rVert_{L^p(W)}$($V\subset\subset W\subset\subset U$):
\begin{equation}
\begin{split}
\int_V|f^\varepsilon(x)|^p \, dx&\leq\int_V\left(\int_{B(x,\varepsilon)} \eta_\varepsilon(x-y) |f(y)|^p\ dy\right)\ dx\\
&\leq\int_W|f(y)|^p \left(\int_{B(y,\varepsilon)}\eta_\varepsilon(x-y)\ dx\right)\ dy=\int_W|f(y)|^p\ dy,
\end{split}
\end{equation}
provided $\varepsilon>0$ is sufficiently small.
It is clear that condition “$\varepsilon\to 0$” is used to make the second ‘ $\leq$’ happen. But I don’t know how. Is Fubini’s theorem or LDC applied?
Edit: Maybe I should post the entire proof to see more clear:
proof of Step 4
Best Answer
Since $V\subset \cup_{x\in V} B(x,\epsilon) \subset W$ if $\epsilon$ small enough, we can prove: $\{(x,y): x\in V , y\in B(x,\epsilon)\} \subset \{(x,y): y \in W, x\in B(y,\epsilon)\}.$
Proof:
$\forall (x,y) \in \{(x,y): x\in V , y\in B(x,\epsilon)\}$, we know $x \in V$, then $y \in B(x,\epsilon)\subset \cup_{x\in V} B(x,\epsilon)\subset W$. And $y \in B(x,\epsilon)$ imples $x\in B(y,\epsilon)$. Therefore $(x,y) \in \{(x,y): y \in W, x\in B(y,\epsilon)\}$, which proves "$\subset$".
Before applying Funibi, we conclude $\int_{ \{(x,y): x\in V , y\in B(x,\epsilon)\}} \le \int_{\{(x,y): y \in W, x\in B(y,\epsilon)\}}$ given the fact that the integrand is nongegative.