Unclear application of Fubini's theorem in Radamacher's theorem proof Let $f: \mathbb{R}^n \mapsto \mathbb{R} $ be a Lipschitz function. Let $v \in \mathbb{R}^n$ be a unit vector, $ x \in \mathbb{R}^n$ and define the directional derivative:
$$ D_{v}f(x)=\lim_{t \rightarrow 0} \frac{f(x+tv)-f(x)}{t} $$
where such limit exists.
Let $A_v=\{ x \in \mathbb{R}^n : \nexists D_{v}f(x)  \}$, we note that $A_v$ is measurable and we want to show that $\lambda^{n}(A_v)=0$. To this end, we define the real function $g$ such that $g(t)=f(x+tv)$ (here $x$ and $v$ are fixed). Since $f$ is Lipschitz, $g$ is also Lipschitz and hence a.e differentiable, concluding that $ \mathcal{H}^{1}(A_v \cap L_v)=0$ for every line $L_v$ which is parallel to $v$.
Then, by Fubini's theorem:
$$ \lambda^n(A_v)=\int_{v^{\perp}} \mathcal{H}^{1}(A_v \cap L_x) \, dx =0 \qquad (*) $$ 
where $L_x$ is the line through $x$ parallel to $v$.
The first part of this argument is clear, but I cannot understand how Fubini's theorem is applied to get the last equality. On the other hand, I think that $(*)$ may be obtained applying the coarea formula (again, I can't see precisely how), but probaably the coarea formula is too much for this one. Any help is appreciated.
 A: Let $w$ be the characteristic function of $A_v$, that is $w=1$ on $A_v$ and $0$ elsewhere. This is a  Borel measurable function. (Verification takes some work, but basically if you start with something Borel measurable, like $f$ is, and consider the set where some limit with $f$ exists, you are still within Borel measurable realm, as the existence of a limit can be expressed in terms of  countable set operations.) The function $w$ is also nonnegative, so Tonelli's theorem applies (Fubuni for nonnegative functions, without the requirement of having finite integral.) 
Write $\mathbb{R}^n$  as the product of a line $L_0$ with direction $v$ and the complementary hyperplane $v^\perp$. Then
$$
\lambda^n(A_v)= \int_{\mathbb{R}^n} w 
= \int_{v^{\perp}} \int_{L_x} w\,dt\,dx = 
\int_{v^{\perp}} \mathcal{H}^{1}(A_v \cap L_x) \, dx
$$
as claimed. 
Usually one skips the step with introducing $w$ and goes straight to: the $n$-dimensional measure is the integral of parallel $k$-dimensional measures of $k$-dimensional slices of the set.
