In Chapter 13 of the book "Lebesgue Integration on Euclidean Spaces" by Frank Jones, Problem 32 asks the reader to prove that if a function $f \in L^{1}(\mathbb{R}^n)$ is such that \begin{eqnarray*} \int_H f(x) \ dx =0 \end{eqnarray*} for every Half Space $H$, where a Half Space is defined as \begin{eqnarray*} H = \{x\in\mathbb{R}^n \ | \ x \cdot \xi \leq c \} \end{eqnarray*} for some fixed $\xi \in\mathbb{R}^n$ and $c \in \mathbb{R}$, then $f =0$ a.e.
The book gave a hint for the problem, and it goes like this ($\hat{f}$ denotes Fourier Transform):
"You should try to prove that $\hat{f}=0$, then apply the Fourier Inversion Theorem. Denote $x^{\prime} = (x_1,\ldots,x_{n-1})$. Why does \begin{eqnarray*} \int^c_{-\infty} dx_n \int_{\mathbb{R}^{n-1}} f(x) \ dx^{\prime} =0 \end{eqnarray*} for every $c$? Why does this imply \begin{eqnarray*} \int_{\mathbb{R}^{n-1}} f(x^{\prime},x_n) \ dx^{\prime}=0 \end{eqnarray*} for a.e. $x_n\in\mathbb{R}$? Why does this imply \begin{eqnarray*} \int_{\mathbb{R}^{n}} f(x) e^{-itx_n}\ dx=0 \end{eqnarray*} for every $t\in\mathbb{R}$?"
I have no problem with the hint, but I have no idea how to proceed from here. The statement only implies that $\hat{f}(x)=0$ for every $x$ of the form $(0,\ldots,t,\ldots,0)$. From here, how can I prove that $\hat{f}(x) =0$ for every $x \in \mathbb{R}^n$? Any help is appreciated, thanks!
For $c \in \mathbb R$ and $\xi \in \mathbb R^n$, let $H(c,\xi) = \{ x \in \mathbb R^n \mid x \cdot \xi \leq c \}.$
Fix $\xi \in \mathbb R^n$. We can then write $$\int_{\mathbb R^n} f(x) \, dx = \frac{1}{|\xi|} \int_{-\infty}^{\infty} \int_{\partial H(c,\xi)} f(x' + t \xi) \, dx' \, dt.$$ and $$\int_{H(c,\xi)} f(x) \, dx = \frac{1}{|\xi|} \int_{-\infty}^{c} \int_{\partial H(c,\xi)} f(x' + t \xi) \, dx' \, dt.$$
Since $\int_{H(c,\xi)} f(x) \, dx = 0$ for all $c$ we have $$0 = \frac{d}{dc} \int_{H(c,\xi)} f(x) \, dx = \frac{1}{|\xi|} \int_{\partial H(c,\xi)} f(x' + c\xi) \, dx'$$ for a.e. $c \in \mathbb R.$
Therefore we have $$e^{-i t |\xi|^2} \int_{\partial H(c,\xi)} f(x' + c \xi) \, dx = 0$$ and thus $$\int_{-\infty}^{\infty} e^{-i t |\xi|^2} \int_{\partial H(c,\xi)} f(x' + c \xi) \, dx \, dt = 0.$$
Since $x = x' + c \xi$ gives $\xi \cdot x = c |\xi|^2$, this can be written as $$\int_{\mathbb R^n} f(x) \, e^{-i \xi \cdot x} \, dx = 0.$$
Now, $\xi$ was arbitrary. Thus the last equation is valid for all $\xi$ so $$\hat f(\xi) = 0$$ and the Fourier inversion formula gives us $f(x) = 0$ for a.e. $x \in \mathbb R^n.$