Let $f$ be a bounded measurable function on $R^n (n \ge 2 )$ such that it satishfies mean value property for a single radius $t >0$ i.e. ${\int_{S^{n-1}}f(x+ty) d\sigma(y)}=f(x)$ where $d\sigma(y)$ is the normalized surface measure of unit sphere. Assume further that $f$ is constant almost everywhere. Then either show that $f$ is constant or provide a counter example.
N.B.- We don't identify two bounded function if they differ by set of measure zero. In my question I have fixed a $f$.