Approximating the Absolutely Continuous Measures Invariant Under General Maps of the Interval

Let $\tau :I \to I$ be a nonsingular, piecewise continuous transformation which admits a unique absolutely continuous invariant measure $\mu$ with density function ${f^ * }$. The main result establishes the fact that ${f^ * }$ can be approximated weakly by the density functions of a sequence of measures invariant under piecewise linear Markov maps $\left \{ {{\tau _n}} \right \}$ which approach $\tau$ uniformly.