Function f such that $\oint_{B} \int_{\mathbb{R}^3} f(x+y) dx dy =0$.

Question

Let $f \in L^1(\mathbb{R}^3)$ such that

$$\oint_{B} \int_{\mathbb{R}^3} f(x+y) dx dy =0$$ for any bounded set $B$ of $\mathbb{R}^3$. I feel like the following is true :

$$\int_{\mathbb{R}^3} f(x) dx =0$$ but I'm having trouble justifying it properly. Does anyone have a proof or a counter-exemple ?

Thanks for the help.

Answer 1

It is actually quite simple. Fix any B, by a change of variable, for any fixed $y \in B$, one has :

$$\int_{\mathbb{R}^3} f(x+y) dx = \int_{\mathbb{R}^3} f(z) dz$$ so one has from the condition

$$\int_{B} \int_{\mathbb{R}^3} f(z) dz = |B|\int_{\mathbb{R}^3} f(z) dz = 0$$ hence the result.