Invariant measures and the compactness of the domain

We consider piecewise monotonic and expanding transformations τ of a real interval (not necessarily bounded) into itself with countable number of points of discontinuity of τ' and with some conditions on the variation ${\displaystyle V_{\begin{array}{cc}0& x\end{array}}\left(\frac{1}{|\tau \prime |}\right)}$ which need not be a bounded function (although it is bounded on any compact interval). We prove that such transformations have absolutely continuous invariant measures. This result generalizes all previous "bounded variation" existence theorems.