  We construct examples of $C^{1}$ Anosov diffeomorphisms on $\mathbb{T}^{2}$
which are of Krieger type ${\rm III}_{1}$ with respect to Lebesgue measure.
This shows that the Gurevic Oseledec phenomena that conservative $C^{1+\alpha}$
Anosov diffeomorphisms have a smooth invariant measure does not hold true in
the $C^{1}$ setting.
