Inequality Involving $\int_V \left(\int_{B(x,\epsilon)} \eta_{\epsilon}(x-y) |f(y)|^p dy\right) dx $ I was reading the book "Partial Differential Equation" written by Lawrence C. Evans, coming up with a question.
On page 718, Evans wrote

$$\int_V \left(\int_{B(x,\epsilon)} \eta_{\epsilon}(x-y) |f(y)|^p dy\right) dx \\ \leq \int_W |f(y)|^p \left(\int_{B(y,\epsilon)} \eta_{\epsilon}(x-y) dx\right) dy$$

Where $V,W,U$ are  open sets, $V\subset\subset W\subset \subset U$, and $\eta_{\epsilon}(x) := \frac{1}{\epsilon^n} \eta(\frac{x}{\epsilon})$, $\eta$ is the standard mollifier, $f \in L_{loc}^p(U)$
I want to ask why the inequality holds. I was trying to prove it so hard but I have no ideas.

Choise some small $\varepsilon>0$ such that $B(x,\varepsilon)\subset V$, we have
$$\int_V\bigg(\int_{B(x,\varepsilon)}\eta_\varepsilon(x-y)|f(y)|^pdy\bigg)dx=\int_V\bigg(\int_V\eta_\varepsilon(x-y)|f(y)|^p~\chi_{B(x,\varepsilon)}(y)~dy\bigg)dx$$
where $\chi_{B(x,\varepsilon)}(y)$ is characteristic function.
Now, using the Fubini's theorem,
$$\int_V\bigg(\int_V\eta_\varepsilon(x-y)|f(y)|^p~\chi_{B(x,\varepsilon)}(y)~dy\bigg)dx=\int_V|f(y)|^p\bigg(\int_V\eta_\varepsilon(x-y)~\chi_{B(x,\varepsilon)}(y)~dx\bigg)dy$$
fixed variable $y$, the characteristic function $\chi_{B(x,\varepsilon)}(y)=1$ only at $x\in B(y,\varepsilon)$, thus
$$\int_V|f(y)|^p\bigg(\int_V\eta_\varepsilon(x-y)~\chi_{B(x,\varepsilon)}(y)~dx\bigg)dy=\int_V|f(y)|^p\bigg(\int_{B(y,\varepsilon)}\eta_\varepsilon(x-y)dx\bigg)dy.$$
Finally, since the fact that the integrand is positive, then
$$\int_V|f(y)|^p\bigg(\int_{B(y,\varepsilon)}\eta_\varepsilon(x-y)dx\bigg)dy\le \int_W|f(y)|^p\bigg(\int_{B(y,\varepsilon)}\eta_\varepsilon(x-y)dx\bigg)dy.$$
