I am reading the notes provided here: https://link.springer.com/content/pdf/10.1007%2F978-1-4419-6055-9.pdf which I have some questions about.
On page 2 of chapter 1, the Radon transform of a function $f: \mathbb{R}^n \to \mathbb{C}$ on a hyperplane $\xi$ is defined as $\hat{f}(\xi) = \int_{\xi} f(x) dm(x)$, where $dm$ is the Euclidean measure on the hyperplane $\xi$.
Later on on page 4, a relationship between Fourier transforms and Radon transform is noted, that $$\tilde{f}(s \omega) = \int_{-\infty}^{\infty} \int_{\langle x, \omega \rangle = r} f(x) e^{-i s \langle x, \omega \rangle} dm(x) dr $$ where $\tilde{f}$ is the Fourier transform of $f$, $\omega$ is a unit vector and $s \in \mathbb{R}$.
My questions:
What exactly is meant by "Euclidean measure on the hyperplane on $\xi$"? The notes do not define it and Google doesn't seem to turn up anything.
I do not understand how the formula for $\tilde{f}(s \omega)$ is obtained. Here, the Fourier transform is written as $\tilde{f}(\omega) = \int f(x) e^{-i \langle x, \omega \rangle} dx$, but I do not see how it can be transformed into the equation involving the Radon Transform.