Proof of weak derivatives in Evans PDE?

Question

In the textbook of Partial differential equation of Evans.

Why from $\int_U(v-\overline v)\phi dx=0$ for all $\phi \in C_c^\infty (U)$, we can get $v-\overline v=0$ a.e.?

How to prove it?

Thanks!

Answer 1

First see $\S$ C.5 of Evans's book.

Now to prove the statement, we simply let $\phi(y)=\eta_\epsilon(x-y)$ and then apply Theorem 7 (ii). The main idea here is Lebesgue's Differentiation Theorem.