Why This Exachange of Integral Works?

Question

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

Answer 1

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.

Answer 2

For positive integrands, many call this Tonelli's Theorem, which doesnt even require integrability to change order of integration. The reason for $ \varepsilon $ small is so that $ B(y, \varepsilon) $ is inside $ W $ for all $ y \in V $.

Answer 3

The diagram is a complement of yuguaw’s answer. The green area is $V_\epsilon=\{(x,y)|x\in V, y\in B(x,\epsilon)\}$ and the blue one is $W_\epsilon=\{(u,y)|y\in W, u\in B(y,\epsilon)\}$.

schematic diagram