I read two proofs of Rademacher’s theorem, on the book Measure Theory and Fine Properties of Functions by Evans, P103 and Sets of Finite Perimeter and Geometric Variation Problems by Francesco, P75.
(1) On Evans’s book, there is a step to derive $$\int_{\mathbb{R}^n} D_v f(x) \zeta(x) \,dx = \int_{\mathbb{R}^n}(v \cdot \operatorname{grad} f(x)) \zeta(x) \, dx$$ I would like to know why here use Fubini’s Theorem and absolute continuity of $f$ on lines.
(2) On Francesco’s book, in the last step, he claims that $$\int_{\mathbb{R}^n} g_0 \nabla \varphi = 0, \forall \varphi \in C_c^\infty(B)$$ I would like to know how to derive this result.
I only have the first book in hand. The author uses Fubini's theorem and absolute continuity in this line to make sure changing order of summation and integral is valid $$\int_{\mathbb{R}^{n}}f\left(x\right)\sum_{i=1}^{n}v_{i}\zeta_{x_{i}}\left(x\right)dx=\sum_{i=1}^{n}v_{i}\int_{\mathbb{R}^{n}}f\left(x\right)\zeta_{x_{i}}\left(x\right)dx,$$ note that summation can be seen as an integral respect to some counting measure $N$, the absolute continuity of $f$ ensures that it can be seen as a Radon–Nikodym derivative of some measure $\mu$ with respect to Lebesgue measure. LHS is integral with respect to $N$ first then $\mu$, RHS is integral with respect to $\mu$ fisrt then $N$, Fubini's theorem applies here.