I am trying to understand the Dennis Kriventsov's note on Tomas-Stein restriction theorem. These are available here:
https://math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Kriventsov.pdf
In page 10, it is said that
$$\int e^{2\pi i (x\cdot \xi+t|\xi|^2)} \hat{f}(\xi) d\xi\quad (1)$$ can be represented as the inverse Fourier transform of $\hat{f}\mu$ where $\mu$ is the measuer on ${\mathbb{R}}^{n}\times \mathbb{R}$ given by $$\int_{\mathbb{R}^{n}\times \mathbb{R}}\phi(\xi,t) d\mu=\int_{\mathbb{R}^{n}}\phi(\xi,|\xi|^2) d\xi,$$ where $\phi$ is continuous. I can not understand this statement. How do we go from the definition of $\mu$ to the fact that $(1)$ is the inverse Fourier transform of $\hat{f}\mu$ ?
The action of the inverse Fourier transform (in $\mathbb{R}^{n+1}$) of the measure $\hat{f}\mu$ on a test function $\phi \in \mathcal{S}(\mathbb{R}^{n+1})$ is
$$\langle (\hat{f}\mu\check) \ , \phi\rangle = \langle \hat f\mu,\check\phi\rangle =\int_{\mathbb{R}^{n+1}} \hat{f}(\xi) \int_{\mathbb{R}^{n+1}}\phi(x,t)e^{2\pi i(x,t)\cdot(\xi,\eta)}dx \ dt \ d\mu(\xi,\eta)\\ = \int_{\mathbb{R}^{n}} \hat{f}(\xi) \int_{\mathbb{R}^{n+1}}\phi(x,t)e^{2\pi i(x,t)\cdot(\xi,|\xi|^2)}dx \ dt \ d\xi\\ =\int_{\mathbb{R}^{n+1}}\phi(x,t)\int_{\mathbb{R}^{n}} \hat{f}(\xi)e^{2\pi i(x\cdot\xi+t|\xi|^2)}d\xi \ dx \ dt\\$$
Which shows that formally
$$(\hat{f}\mu\check) = \int_{\mathbb{R}^{n}} \hat{f}(\xi)e^{2\pi i(x\cdot\xi+t|\xi|^2)}d\xi$$