Let $f \in L^1(\mathbb{R}^d)$ be a function such that\begin{align*}\int_{B_1(x_0)} f(x)\,dx = 0\end{align*}for every $x_0 \in \mathbb{R}^d$ (where $B_1$ denotes the unit ball). I want to prove that $f = 0$ a.e. This should be easy in theory (it was left as an exercise to some undergraduate friends of mine), but I wasn't able to find a simple solution. Some considerations:
- (a) The $f \in L^1(\mathbb{R}^d)$ assumption has to play a role, there are counterexamples when $f$ is simply $L^1_{loc}$.
- (b) The case $d=1$ is indeed straightforward, one can simply consider the difference $\int_{B_1(x)} f - \int_{B_1(y)} f$ and obtain that f must be periodic.
- (c) One could apply the Fourier transform to $f*\chi_{B_1}$ and obtain that $\mathcal{F}(f) \mathcal{F}(\chi_{B_1})=0$. This should imply $\mathcal{F}(f)=0$ (and thus $f=0$), but I would prefer not to deal with Bessel functions and their zeroes (moreover, in theory my friends don't know yet what a Fourier transform actually is).
Any idea? For reference, assume that the solution should only make use of basic properties of the convolution and $L^p$ spaces.