  In this paper, we prove a threshold result on the existence of a circularly
invariant uniformizable probability measure (CIUPM) for linear transformations
with non-zero slope on the line. We show that there is a threshold constant $c$
depending only on the slope of the linear transformation such that there exists
a CIUPM if and only if its support has a diameter at least as large as $c.$
Moreover, the CIUPM is unique up to translation if the diameter of the support
equals $c.$
