Reference for an integral formula Good morning, 
I'm reading a paper of W. Stoll in which the author uses some implicit facts (i.e. he states them without proofs and references) in measure theory. So I would like to ask the following question:
Let $G$ be a bounded domain in $\mathbb{R}^n$ and $S^{n-1}$ the unit sphere in $\mathbb{R}^n.$ For each $a\in S^{n-1},$ define $L(a) = \{x.a~:~ x\in \mathbb{R}\}.$ Denote by $L^n$ the n-dimensional Lebesgue area. Is the following formula true? $$\int_{a\in S^{n-1}}L^1(G\cap L(a)) = L^n(G) = \mathrm{vol}(G).$$
Could anyone please show me a reference where there is a proof for this? If this formula is not true, how will we correct it?
Thanks in advance,
Duc Anh

This formula seems to be false. Consider the case of the unit disk in $\mathbb{R}^2$, $D^2$. This is obviously bounded. 
$L^1(D^2 \cap L(a)) = 2$ for any $a$ in $S^1$, as the radius of $D^2$ is 1, and the intersection of the line through the origin that goes through $a$ and $D^2$ has length 2. The integral on the left is therefore equal to $4\pi$ and $L^2(D^2)$ was known by the greeks to be $\pi$. 
So your formula is like computing an integral in polar coordinates without multiplying the integrand by the determinant of the jacobian. 
