Suppose that the limit \begin{equation*} \text{p.v.}\int_{\mathbb{R}^n}\frac{\Omega(x-y)}{|x-y|^n}f(y)dy \end{equation*} exists for almost every $x\in \mathbb{R}$ for all $f \in C^\infty_c(\mathbb{R}^n)$. Here, $\Omega \in L^1(S^{n-1})$ and for any $\lambda >0$, $\Omega(\lambda x) = \lambda \Omega(x)$. Show that $\int_{S^{n-1}}\Omega d\sigma=0$.
I try to express the integral as $$ \int_{S^{n-1}} \Omega(y') \int_0^\infty f(x-ry')\frac{dr}{r} d\sigma(y') $$ and use some particular choice of $f$ to arrive at a contradcition if the cancellation property $\int_{S^{n-1}}\Omega d\sigma=0$ is not satisfied.