  Suppose $\Lambda$ is a discrete infinite set of nonnegative real numbers. We
say that $ {\Lambda}$ is type $2$ if the series
$s(x)=\sum_{\lambda\in\Lambda}f(x+\lambda)$ does not satisfy a zero-one law.
This means that we can find a non-negative measurable "witness function"
  $f: {\mathbb R}\to [0,+ {\infty})$ such that both the convergence set $C(f,
{\Lambda})=\{x: s(x)<+ {\infty} \}$ and its complement the divergence set $D(f,
{\Lambda})=\{x: s(x)=+ {\infty} \}$ are of positive Lebesgue measure. If $
{\Lambda}$ is not type $2$ we say that $ {\Lambda}$ is type $1$.
  The main result of our paper answers a question raised by Z. Buczolich, J-P.
Kahane, and D. Mauldin. By a random construction we show that one can always
choose a witness function which is the characteristic function of a measurable
set.
  We also consider the effect on the type of a set $ {\Lambda}$ if we randomly
delete its elements.
  Motivated by results concerning weighted sums $\sum c_n f(nx)$ and the
Khinchin conjecture, we also discuss some results about weighted sums
  $\sum_{n=1}^{\infty}c_n f(x+\lambda_n)$.
