Reference for an integral formula

Question

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

Answer 1

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.