  We show that if points of supports of two discrete "not very thick" Fourier
transformable measures on LCA groups tend to one another at infinity and the
same is true for the masses at these points, then these measures coincide. The
result is valid for discrete almost periodic measures on LCA groups too. Also,
we show that the result is false for some discrete "thick" measures. To do
this, we construct a discrete almost periodic measure on the real axis, whose
masses at the points of support tend to zero as these points approach infinity.
