  We construct inhomogenous Markov measures for which the shift is of Kreiger
type ${\rm III}_{1}$. These measures are fully supported on a toplogical markov
shift space of the hyperbolic toral automorphism
$f(x,y)=\left(\{x+y\},x\right):\mathbb{T}^{2}\to\mathbb{T}^{2}$ and are used in
a subsequent paper in the construction of $C^{1}$ Anosov diffeomorphisms of
$\mathbb{T}^{2}$ without an absolutely continuous invariant measure.
