Every translation of $f$ has no radial part 
Let $f\in L^1(\mathbb{R}^n)$. Define the radial part of $f$ as
  $$f_0(x)=\int_{S^{n-1}} f(||x||\omega)\,d\omega$$ where $\,d\omega$ is
  the normalised surface integral over $S^{n-1}$. Define translation of
  $f$ as $\ell_xf(y)=f(y-x)$. Suppose that $(\ell_xf)_0\equiv 0$ for all
  $x\in\mathbb{R}^n$. Then show that $f\equiv 0$.

I can prove it for continuous functions. But $f\in L^1$ is creating problem. Any hint/suggestion is welcomed.

Since the integral over every sphere is zero, it follows (by Fubini) that the integral over every ball is zero. Hence, $f=0$ at every Lebesgue point, which is a.e. point (see the Lebesgue differentiation theorem).
