  We show that there is a sharp threshold in dimension one for the transport
cost between the Lebesgue measure $\lambda$ and an invariant random measure
$\mu$ of unit intensity to be finite. We show that for \emph{any} such random
measure the $L^1$ cost are infinite provided that the first central moments
$\mathbb{E}[|n-\mu([0,n))|]$ diverge. Furthermore, we establish simple and
sharp criteria, based on the variance of $\mu([0,n)]$, for the $L^p$ cost to be
finite for $0<p<1$.
