Description of Pompeiu sets in terms of approximations of their indicator functions

Let $H$ be an open upper half-space in $mathbb R^n$, $ngeq2$, and assume that $A$ is a non-empty, open, bounded subset of $mathbb R^n$ such that $overline{A}subset H$ and the exterior of $A$ is connected. Let $pin[2, +infty).$ It is proved that there is a nonzero function with zero integrals over all sets in $mathbb R^n$ congruent to $A$ if and only if the indicator function of $A$ is the limit in $L^p(H)$ of a sequence of linear combinations of indicator functions of balls in $H$ with radii proportional to positive zeros of the Bessel function $J_{n/2}$. The proportionality coefficient here is the same for all balls and depends only on $A$.